# Real Limits, Continuity, Derivatives, and Polynomial Integration: Seven-Round Workflow 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. - Stage 1 · required ## [Sequence](/en/mko/mko-sequence/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-sequence) [Concept guides](/en/derivations/?q=mko-sequence) - Stage 1 · required ## [Tends-to relation](/en/mko/mko-tends-to-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-tends-to-relation) [Concept guides](/en/derivations/?q=mko-tends-to-relation) - Stage 1 · required ## [Quantifiers and explicit domains](/en/mko/mko-quantifier/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-quantifier) [Concept guides](/en/derivations/?q=mko-quantifier) - Stage 1 · required ## [Rational number](/en/mko/mko-rational-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-rational-number) [Concept guides](/en/derivations/?q=mko-rational-number) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Absolute value](/en/mko/mko-absolute-value/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-absolute-value) [Concept guides](/en/derivations/?q=mko-absolute-value) - Stage 1 · required ## [Polynomial factorization over the rationals](/en/mko/mko-factorization/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-factorization) [Concept guides](/en/derivations/?q=mko-factorization) - Stage 1 · required ## [Average rate of change between distinct inputs](/en/mko/mko-rate-of-change/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-rate-of-change) [Concept guides](/en/derivations/?q=mko-rate-of-change) - Stage 1 · required ## [Univariate polynomials over the rationals](/en/mko/mko-polynomial/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-polynomial) [Concept guides](/en/derivations/?q=mko-polynomial) - Stage 1 · required ## [Function composition](/en/mko/mko-composition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-composition) [Concept guides](/en/derivations/?q=mko-composition) - Stage 1 · required ## [Exact area of half-open rectangles and unit-cell unions](/en/mko/mko-area/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-area) [Concept guides](/en/derivations/?q=mko-area) - Stage 2 · optional ## [Sequence limits and exact tail cutoffs](/en/mko/mko-sequence-limit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) 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. [Related questions](/en/explore/?mko=mko-sequence-limit) [Concept guides](/en/derivations/?q=mko-sequence-limit) - Stage 3 · optional ## [Function limits on punctured real domains](/en/mko/mko-function-limit/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) An ε–δ limit controls values near a while excluding a itself; the center value is not part of the punctured-limit condition. [Related questions](/en/explore/?mko=mko-function-limit) [Concept guides](/en/derivations/?q=mko-function-limit) - Stage 4 · optional ## [Point values and continuity of real functions](/en/mko/mko-continuity/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Continuity on the whole real domain requires a defined center value agreeing with the two-sided limit; extension and original-function continuity are distinct. [Related questions](/en/explore/?mko=mko-continuity) [Concept guides](/en/derivations/?q=mko-continuity) - Stage 5 · optional ## [Real derivatives at a point and finite difference quotients](/en/mko/mko-derivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) A derivative is the two-sided limit of nonzero-step quotients, not any one finite-step quotient; an absolute-value corner retains nonexistence. [Related questions](/en/explore/?mko=mko-derivative) [Concept guides](/en/derivations/?q=mko-derivative) - Stage 6 · optional ## [Differentiating polynomial sums, products, and compositions](/en/mko/mko-derivative-rules/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) Four differentiation rules retain differentiability premises; this pulse evaluates them exactly for finite integer-coefficient polynomials. [Related questions](/en/explore/?mko=mko-derivative-rules) [Concept guides](/en/derivations/?q=mko-derivative-rules) - Stage 7 · optional ## [Polynomial antiderivatives and the choice of constant](/en/mko/mko-antiderivative/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) An antiderivative differentiates to f on its declared interval; this pulse selects a zero-constant rational-coefficient representative for full-real polynomials. [Related questions](/en/explore/?mko=mko-antiderivative) [Concept guides](/en/derivations/?q=mko-antiderivative) - Stage 8 · optional ## [Signed polynomial definite integrals on finite intervals](/en/mko/mko-definite-integral/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261008%2F) 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. [Related questions](/en/explore/?mko=mko-definite-integral) [Concept guides](/en/derivations/?q=mko-definite-integral) Path source & statuspath-pulse-foundations-20261008 · candidate [Original paths JSON](/data/architecture/learning-paths.json)