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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261007-advance-20261008-r03-continuity-foundations · 2026-10-07 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
On the whole real domain, f(x)=(0)x+(0). Find the defined point value f(0).
0
Exact substitution gives m*a+b=0. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(-2)x+(-11). Find the defined point value f(0).
-11
Exact substitution gives m*a+b=-11. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(3)x+(-10). Find the defined point value f(3).
-1
Exact substitution gives m*a+b=-1. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(-4)x+(-9). Find the defined point value f(4).
-25
Exact substitution gives m*a+b=-25. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(5)x+(-8). Find the defined point value f(1).
-3
Exact substitution gives m*a+b=-3. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(-6)x+(-7). Find the defined point value f(2).
-19
Exact substitution gives m*a+b=-19. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(7)x+(-6). Find the defined point value f(3).
15
Exact substitution gives m*a+b=15. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(-8)x+(-5). Find the defined point value f(4).
-37
Exact substitution gives m*a+b=-37. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(9)x+(-4). Find the defined point value f(1).
5
Exact substitution gives m*a+b=5. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(-10)x+(-3). Find the defined point value f(2).
-23
Exact substitution gives m*a+b=-23. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(11)x+(-2). Find the defined point value f(3).
31
Exact substitution gives m*a+b=31. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(-12)x+(-1). Find the defined point value f(4).
-49
Exact substitution gives m*a+b=-49. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(13)x+(0). Find the defined point value f(1).
13
Exact substitution gives m*a+b=13. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(-14)x+(1). Find the defined point value f(2).
-27
Exact substitution gives m*a+b=-27. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(15)x+(2). Find the defined point value f(3).
47
Exact substitution gives m*a+b=47. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(-16)x+(3). Find the defined point value f(4).
-61
Exact substitution gives m*a+b=-61. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(17)x+(4). Find the defined point value f(1).
21
Exact substitution gives m*a+b=21. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(-18)x+(5). Find the defined point value f(2).
-31
Exact substitution gives m*a+b=-31. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(19)x+(6). Find the defined point value f(3).
63
Exact substitution gives m*a+b=63. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(-20)x+(7). Find the defined point value f(4).
-73
Exact substitution gives m*a+b=-73. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(21)x+(8). Find the defined point value f(1).
29
Exact substitution gives m*a+b=29. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(-22)x+(9). Find the defined point value f(2).
-35
Exact substitution gives m*a+b=-35. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(23)x+(10). Find the defined point value f(3).
79
Exact substitution gives m*a+b=79. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(-24)x+(11). Find the defined point value f(4).
-85
Exact substitution gives m*a+b=-85. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
On the whole real domain, f(x)=(25)x+(12). Find the defined point value f(1).
37
Exact substitution gives m*a+b=37. This is the stated total affine model's point value, not a silently filled value of a punctured function.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-affine-value · candidate
The function is defined on all R. The left branch at x<-12 is (1)x+(-1); the right branch at x>-12 is (-1)x+(-25). Its explicit center value is f(-12)=-13. Is it continuous at -12?
Yes
The side limits are [-13, -13], and the center value is -13. All three must agree, giving Yes; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<-11 is (2)x+(24); the right branch at x>-11 is (-2)x+(-20). Its explicit center value is f(-11)=3. Is it continuous at -11?
No
The side limits are [2, 2], and the center value is 3. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<-10 is (3)x+(29); the right branch at x>-10 is (-3)x+(-29). Its explicit center value is f(-10)=0. Is it continuous at -10?
No
The side limits are [-1, 1], and the center value is 0. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<-9 is (4)x+(26); the right branch at x>-9 is (-1)x+(-19). Its explicit center value is f(-9)=-10. Is it continuous at -9?
Yes
The side limits are [-10, -10], and the center value is -10. All three must agree, giving Yes; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<-8 is (1)x+(0); the right branch at x>-8 is (-2)x+(-24). Its explicit center value is f(-8)=-9. Is it continuous at -8?
No
The side limits are [-8, -8], and the center value is -9. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<-7 is (2)x+(6); the right branch at x>-7 is (-3)x+(-27). Its explicit center value is f(-7)=-8. Is it continuous at -7?
No
The side limits are [-8, -6], and the center value is -8. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<-6 is (3)x+(11); the right branch at x>-6 is (-1)x+(-13). Its explicit center value is f(-6)=-7. Is it continuous at -6?
Yes
The side limits are [-7, -7], and the center value is -7. All three must agree, giving Yes; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<-5 is (4)x+(15); the right branch at x>-5 is (-2)x+(-15). Its explicit center value is f(-5)=-6. Is it continuous at -5?
No
The side limits are [-5, -5], and the center value is -6. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<-4 is (1)x+(-1); the right branch at x>-4 is (-3)x+(-15). Its explicit center value is f(-4)=-5. Is it continuous at -4?
No
The side limits are [-5, -3], and the center value is -5. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<-3 is (2)x+(2); the right branch at x>-3 is (-1)x+(-7). Its explicit center value is f(-3)=-4. Is it continuous at -3?
Yes
The side limits are [-4, -4], and the center value is -4. All three must agree, giving Yes; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<-2 is (3)x+(4); the right branch at x>-2 is (-2)x+(-6). Its explicit center value is f(-2)=-3. Is it continuous at -2?
No
The side limits are [-2, -2], and the center value is -3. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<-1 is (4)x+(2); the right branch at x>-1 is (-3)x+(-3). Its explicit center value is f(-1)=-2. Is it continuous at -1?
No
The side limits are [-2, 0], and the center value is -2. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<0 is (1)x+(-1); the right branch at x>0 is (-1)x+(-1). Its explicit center value is f(0)=-1. Is it continuous at 0?
Yes
The side limits are [-1, -1], and the center value is -1. All three must agree, giving Yes; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<1 is (2)x+(-1); the right branch at x>1 is (-2)x+(3). Its explicit center value is f(1)=0. Is it continuous at 1?
No
The side limits are [1, 1], and the center value is 0. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<2 is (3)x+(-5); the right branch at x>2 is (-3)x+(9). Its explicit center value is f(2)=1. Is it continuous at 2?
No
The side limits are [1, 3], and the center value is 1. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<3 is (4)x+(-10); the right branch at x>3 is (-1)x+(5). Its explicit center value is f(3)=2. Is it continuous at 3?
Yes
The side limits are [2, 2], and the center value is 2. All three must agree, giving Yes; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<4 is (1)x+(0); the right branch at x>4 is (-2)x+(12). Its explicit center value is f(4)=3. Is it continuous at 4?
No
The side limits are [4, 4], and the center value is 3. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<5 is (2)x+(-6); the right branch at x>5 is (-3)x+(21). Its explicit center value is f(5)=4. Is it continuous at 5?
No
The side limits are [4, 6], and the center value is 4. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<6 is (3)x+(-13); the right branch at x>6 is (-1)x+(11). Its explicit center value is f(6)=5. Is it continuous at 6?
Yes
The side limits are [5, 5], and the center value is 5. All three must agree, giving Yes; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<7 is (4)x+(-21); the right branch at x>7 is (-2)x+(21). Its explicit center value is f(7)=6. Is it continuous at 7?
No
The side limits are [7, 7], and the center value is 6. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<8 is (1)x+(-1); the right branch at x>8 is (-3)x+(33). Its explicit center value is f(8)=7. Is it continuous at 8?
No
The side limits are [7, 9], and the center value is 7. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<9 is (2)x+(-10); the right branch at x>9 is (-1)x+(17). Its explicit center value is f(9)=8. Is it continuous at 9?
Yes
The side limits are [8, 8], and the center value is 8. All three must agree, giving Yes; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<10 is (3)x+(-20); the right branch at x>10 is (-2)x+(30). Its explicit center value is f(10)=9. Is it continuous at 10?
No
The side limits are [10, 10], and the center value is 9. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<11 is (4)x+(-34); the right branch at x>11 is (-3)x+(45). Its explicit center value is f(11)=10. Is it continuous at 11?
No
The side limits are [10, 12], and the center value is 10. All three must agree, giving No; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The function is defined on all R. The left branch at x<12 is (1)x+(-1); the right branch at x>12 is (-1)x+(23). Its explicit center value is f(12)=11. Is it continuous at 12?
Yes
The side limits are [11, 11], and the center value is 11. All three must agree, giving Yes; agreement of the sides alone is insufficient.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-piecewise-point · candidate
The original quotient (x²−(-12)²)/(x−(-12)) is defined only away from -12. Choose the center value of a new continuous extension to all R.
-24
The punctured limit is 2a=-24, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(-11)²)/(x−(-11)) is defined only away from -11. Choose the center value of a new continuous extension to all R.
-22
The punctured limit is 2a=-22, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(-10)²)/(x−(-10)) is defined only away from -10. Choose the center value of a new continuous extension to all R.
-20
The punctured limit is 2a=-20, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(-9)²)/(x−(-9)) is defined only away from -9. Choose the center value of a new continuous extension to all R.
-18
The punctured limit is 2a=-18, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(-8)²)/(x−(-8)) is defined only away from -8. Choose the center value of a new continuous extension to all R.
-16
The punctured limit is 2a=-16, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(-7)²)/(x−(-7)) is defined only away from -7. Choose the center value of a new continuous extension to all R.
-14
The punctured limit is 2a=-14, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(-6)²)/(x−(-6)) is defined only away from -6. Choose the center value of a new continuous extension to all R.
-12
The punctured limit is 2a=-12, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(-5)²)/(x−(-5)) is defined only away from -5. Choose the center value of a new continuous extension to all R.
-10
The punctured limit is 2a=-10, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(-4)²)/(x−(-4)) is defined only away from -4. Choose the center value of a new continuous extension to all R.
-8
The punctured limit is 2a=-8, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(-3)²)/(x−(-3)) is defined only away from -3. Choose the center value of a new continuous extension to all R.
-6
The punctured limit is 2a=-6, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(-2)²)/(x−(-2)) is defined only away from -2. Choose the center value of a new continuous extension to all R.
-4
The punctured limit is 2a=-4, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(-1)²)/(x−(-1)) is defined only away from -1. Choose the center value of a new continuous extension to all R.
-2
The punctured limit is 2a=-2, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(0)²)/(x−(0)) is defined only away from 0. Choose the center value of a new continuous extension to all R.
0
The punctured limit is 2a=0, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(1)²)/(x−(1)) is defined only away from 1. Choose the center value of a new continuous extension to all R.
2
The punctured limit is 2a=2, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(2)²)/(x−(2)) is defined only away from 2. Choose the center value of a new continuous extension to all R.
4
The punctured limit is 2a=4, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(3)²)/(x−(3)) is defined only away from 3. Choose the center value of a new continuous extension to all R.
6
The punctured limit is 2a=6, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(4)²)/(x−(4)) is defined only away from 4. Choose the center value of a new continuous extension to all R.
8
The punctured limit is 2a=8, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(5)²)/(x−(5)) is defined only away from 5. Choose the center value of a new continuous extension to all R.
10
The punctured limit is 2a=10, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(6)²)/(x−(6)) is defined only away from 6. Choose the center value of a new continuous extension to all R.
12
The punctured limit is 2a=12, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(7)²)/(x−(7)) is defined only away from 7. Choose the center value of a new continuous extension to all R.
14
The punctured limit is 2a=14, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(8)²)/(x−(8)) is defined only away from 8. Choose the center value of a new continuous extension to all R.
16
The punctured limit is 2a=16, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(9)²)/(x−(9)) is defined only away from 9. Choose the center value of a new continuous extension to all R.
18
The punctured limit is 2a=18, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(10)²)/(x−(10)) is defined only away from 10. Choose the center value of a new continuous extension to all R.
20
The punctured limit is 2a=20, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(11)²)/(x−(11)) is defined only away from 11. Choose the center value of a new continuous extension to all R.
22
The punctured limit is 2a=22, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
The original quotient (x²−(12)²)/(x−(12)) is defined only away from 12. Choose the center value of a new continuous extension to all R.
24
The punctured limit is 2a=24, the unique required extension value. The original quotient remains undefined at the center until a separate extension is specified.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-removable-extension · candidate
On all R, The left branch at x<-12 is (1)x+(0); the right branch at x>-12 is (-1)x+(-24). The center value is -12. Give (left limit−center value, right limit−center value).
(0, 0)
The side limits are -12 and -12; subtract the independently given center value to obtain (0, 0). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<-11 is (2)x+(12); the right branch at x>-11 is (-2)x+(-34). The center value is -11. Give (left limit−center value, right limit−center value).
(1, -1)
The side limits are -10 and -12; subtract the independently given center value to obtain (1, -1). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<-10 is (3)x+(22); the right branch at x>-10 is (-3)x+(-37). The center value is -10. Give (left limit−center value, right limit−center value).
(2, 3)
The side limits are -8 and -7; subtract the independently given center value to obtain (2, 3). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<-9 is (4)x+(30); the right branch at x>-9 is (-1)x+(-21). The center value is -9. Give (left limit−center value, right limit−center value).
(3, -3)
The side limits are -6 and -12; subtract the independently given center value to obtain (3, -3). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<-8 is (1)x+(4); the right branch at x>-8 is (-2)x+(-19). The center value is -8. Give (left limit−center value, right limit−center value).
(4, 5)
The side limits are -4 and -3; subtract the independently given center value to obtain (4, 5). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<-7 is (2)x+(12); the right branch at x>-7 is (-3)x+(-33). The center value is -7. Give (left limit−center value, right limit−center value).
(5, -5)
The side limits are -2 and -12; subtract the independently given center value to obtain (5, -5). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<-6 is (3)x+(18); the right branch at x>-6 is (-1)x+(-5). The center value is -6. Give (left limit−center value, right limit−center value).
(6, 7)
The side limits are 0 and 1; subtract the independently given center value to obtain (6, 7). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<-5 is (4)x+(22); the right branch at x>-5 is (-2)x+(-22). The center value is -5. Give (left limit−center value, right limit−center value).
(7, -7)
The side limits are 2 and -12; subtract the independently given center value to obtain (7, -7). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<-4 is (1)x+(8); the right branch at x>-4 is (-3)x+(-7). The center value is -4. Give (left limit−center value, right limit−center value).
(8, 9)
The side limits are 4 and 5; subtract the independently given center value to obtain (8, 9). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<-3 is (2)x+(12); the right branch at x>-3 is (-1)x+(-15). The center value is -3. Give (left limit−center value, right limit−center value).
(9, -9)
The side limits are 6 and -12; subtract the independently given center value to obtain (9, -9). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<-2 is (3)x+(14); the right branch at x>-2 is (-2)x+(5). The center value is -2. Give (left limit−center value, right limit−center value).
(10, 11)
The side limits are 8 and 9; subtract the independently given center value to obtain (10, 11). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<-1 is (4)x+(14); the right branch at x>-1 is (-3)x+(-15). The center value is -1. Give (left limit−center value, right limit−center value).
(11, -11)
The side limits are 10 and -12; subtract the independently given center value to obtain (11, -11). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<0 is (1)x+(12); the right branch at x>0 is (-1)x+(13). The center value is 0. Give (left limit−center value, right limit−center value).
(12, 13)
The side limits are 12 and 13; subtract the independently given center value to obtain (12, 13). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<1 is (2)x+(12); the right branch at x>1 is (-2)x+(-10). The center value is 1. Give (left limit−center value, right limit−center value).
(13, -13)
The side limits are 14 and -12; subtract the independently given center value to obtain (13, -13). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<2 is (3)x+(10); the right branch at x>2 is (-3)x+(23). The center value is 2. Give (left limit−center value, right limit−center value).
(14, 15)
The side limits are 16 and 17; subtract the independently given center value to obtain (14, 15). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<3 is (4)x+(6); the right branch at x>3 is (-1)x+(-9). The center value is 3. Give (left limit−center value, right limit−center value).
(15, -15)
The side limits are 18 and -12; subtract the independently given center value to obtain (15, -15). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<4 is (1)x+(16); the right branch at x>4 is (-2)x+(29). The center value is 4. Give (left limit−center value, right limit−center value).
(16, 17)
The side limits are 20 and 21; subtract the independently given center value to obtain (16, 17). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<5 is (2)x+(12); the right branch at x>5 is (-3)x+(3). The center value is 5. Give (left limit−center value, right limit−center value).
(17, -17)
The side limits are 22 and -12; subtract the independently given center value to obtain (17, -17). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<6 is (3)x+(6); the right branch at x>6 is (-1)x+(31). The center value is 6. Give (left limit−center value, right limit−center value).
(18, 19)
The side limits are 24 and 25; subtract the independently given center value to obtain (18, 19). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<7 is (4)x+(-2); the right branch at x>7 is (-2)x+(2). The center value is 7. Give (left limit−center value, right limit−center value).
(19, -19)
The side limits are 26 and -12; subtract the independently given center value to obtain (19, -19). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<8 is (1)x+(20); the right branch at x>8 is (-3)x+(53). The center value is 8. Give (left limit−center value, right limit−center value).
(20, 21)
The side limits are 28 and 29; subtract the independently given center value to obtain (20, 21). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<9 is (2)x+(12); the right branch at x>9 is (-1)x+(-3). The center value is 9. Give (left limit−center value, right limit−center value).
(21, -21)
The side limits are 30 and -12; subtract the independently given center value to obtain (21, -21). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<10 is (3)x+(2); the right branch at x>10 is (-2)x+(53). The center value is 10. Give (left limit−center value, right limit−center value).
(22, 23)
The side limits are 32 and 33; subtract the independently given center value to obtain (22, 23). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<11 is (4)x+(-10); the right branch at x>11 is (-3)x+(21). The center value is 11. Give (left limit−center value, right limit−center value).
(23, -23)
The side limits are 34 and -12; subtract the independently given center value to obtain (23, -23). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
On all R, The left branch at x<12 is (1)x+(24); the right branch at x>12 is (-1)x+(49). The center value is 12. Give (left limit−center value, right limit−center value).
(24, 25)
The side limits are 36 and 37; subtract the independently given center value to obtain (24, 25). Both defects must be zero for continuity in this model.
Reviewed step-by-step derivation
Point values and continuity of real functions · qf-continuity-point-defects · candidate
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