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Function limits on punctured real domains
An ε–δ limit controls values near a while excluding a itself; the center value is not part of the punctured-limit condition.
OCME · Bilingual mathematics & derivations
Knowledge objects
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
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Point values and continuity of real functions
Continuity on the whole real domain requires a defined center value agreeing with the two-sided limit; extension and original-function continuity are distinct.
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Real derivatives at a point and finite difference quotients
A derivative is the two-sided limit of nonzero-step quotients, not any one finite-step quotient; an absolute-value corner retains nonexistence.
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Differentiating polynomial sums, products, and compositions
Four differentiation rules retain differentiability premises; this pulse evaluates them exactly for finite integer-coefficient polynomials.
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Polynomial antiderivatives and the choice of constant
An antiderivative differentiates to f on its declared interval; this pulse selects a zero-constant rational-coefficient representative for full-real polynomials.
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Signed polynomial definite integrals on finite intervals
Continuous polynomials have oriented definite integrals with exact moment formulas; a finite left Riemann sum is not a general integral or a completed general FTC bridge.
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Fundamental theorem of calculus: continuity and oriented endpoints
With continuity on all R, differentiation of the accumulating integral recovers the integrand; an eligible antiderivative gives an oriented endpoint difference.
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Nonempty finite trees: connectivity, cycles, and unique paths
A finite simple undirected graph is a tree only when nonempty, connected, and acyclic; retain singleton, degree-one leaf, and new-edge cycle conventions.
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Topological space
A topological space consists of an underlying set and a family of open sets containing the empty set and whole space and closed under arbitrary unions and finite intersections.
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Natural-number divisibility
A natural number a divides b when a natural multiplier k satisfies b=a·k; the two roles of zero must be distinguished.
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Greatest common divisor and coprimality
The natural number gcd(a,b) divides both a and b and is divisible by every common divisor; the inputs are coprime when their gcd is 1.
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Prime natural number
A prime natural number is at least 2 and has no positive divisors other than 1 and itself.
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Natural-number congruence modulo a positive modulus
For a fixed positive modulus n, two natural numbers a and b are congruent modulo n when they have the same remainder on division by n.
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Decision problem
A decision problem specifies an input set X and a yes-instance subset Y⊆X; a decider must terminate on every legal input and return the correct Boolean answer.
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Formal statements in a proof assistant
A formal statement is a core term of type Prop obtained by successful elaboration in a fixed version, import environment and local context.
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Open and closed sets
In a fixed topological space, open sets belong to the topology and closed sets have open complements; one set can be both open and closed.
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Neighborhood
A set N is a neighborhood of x exactly when N contains an open set containing x; N itself need not be open.
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Continuous maps between topological spaces
A map is topologically continuous when every open set in its target has an open preimage in its source.
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Abstract computational model
A finite-state computational model describes possible executions through its state set, initial-state set, step relation and halt-state set.
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Algorithm correctness
Algorithm correctness separates partial correctness from termination; total correctness requires both.
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Asymptotic resource bounds
Asymptotic upper, lower and tight bounds require an input-size convention, cost model and quantification over a full tail.
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Proof term
A proof term t has propositional type P in a fixed formal environment and context, and is checked by the kernel.
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Discrete-time dynamical system
A discrete-time dynamical system uses a state space, a self-map and natural-number time to describe stepwise evolution; a finite orbit check does not establish general stability.
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Ordinary differential equation
An ordinary differential equation relates an unknown single-variable function and finitely many derivatives; candidate residual checks are not existence or uniqueness theorems.