OCME · Bilingual mathematics & derivations
Real Limits, Continuity, Derivatives, and Polynomial Integration: Seven-Round Workflow
On this page
- Sequence
- Tends-to relation
- Quantifiers and explicit domains
- Rational number
- Function mapping
- Absolute value
- Polynomial factorization over the rationals
- Average rate of change between distinct inputs
- Univariate polynomials over the rationals
- Function composition
- Exact area of half-open rectangles and unit-cell unions
- Sequence limits and exact tail cutoffs
- Function limits on punctured real domains
- Point values and continuity of real functions
- Real derivatives at a point and finite difference quotients
- Differentiating polynomial sums, products, and compositions
- Polynomial antiderivatives and the choice of constant
- Signed polynomial definite integrals on finite intervals
Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations.
This is a suggested learning sequence, not a universal logical dependency or a formal proof.
Sequence
Existing canonical prerequisite for the new objects in this pulse.
Tends-to relation
Existing canonical prerequisite for the new objects in this pulse.
Quantifiers and explicit domains
Existing canonical prerequisite for the new objects in this pulse.
Rational number
Existing canonical prerequisite for the new objects in this pulse.
Function mapping
Existing canonical prerequisite for the new objects in this pulse.
Absolute value
Existing canonical prerequisite for the new objects in this pulse.
Polynomial factorization over the rationals
Existing canonical prerequisite for the new objects in this pulse.
Average rate of change between distinct inputs
Existing canonical prerequisite for the new objects in this pulse.
Univariate polynomials over the rationals
Existing canonical prerequisite for the new objects in this pulse.
Function composition
Existing canonical prerequisite for the new objects in this pulse.
Exact area of half-open rectangles and unit-cell unions
Existing canonical prerequisite for the new objects in this pulse.
Sequence limits and exact tail cutoffs
A zero-based real sequence converges when every precision admits a natural cutoff controlling the entire tail; this pulse treats an exact reciprocal-tail model.
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.
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.
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.
Differentiating polynomial sums, products, and compositions
Four differentiation rules retain differentiability premises; this pulse evaluates them exactly for finite integer-coefficient polynomials.
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.
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.
Path source & status
path-pulse-foundations-20261008 · candidate