# Knowledge objects Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence. [← Previous](/en/knowledge/?page=3#results)[1](/en/knowledge/?page=1#results) … [3](/en/knowledge/?page=3#results) 4 [5](/en/knowledge/?page=5#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=5#results) 302 results · Page 4 of 13 Knowledge object ## [Function limits on punctured real domains](/en/mko/mko-function-limit/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) An ε–δ limit controls values near a while excluding a itself; the center value is not part of the punctured-limit condition. mko-function-limit · Analysis Knowledge object ## [Point values and continuity of real functions](/en/mko/mko-continuity/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) Continuity on the whole real domain requires a defined center value agreeing with the two-sided limit; extension and original-function continuity are distinct. mko-continuity · Analysis Knowledge object ## [Real derivatives at a point and finite difference quotients](/en/mko/mko-derivative/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) A derivative is the two-sided limit of nonzero-step quotients, not any one finite-step quotient; an absolute-value corner retains nonexistence. mko-derivative · Analysis Knowledge object ## [Differentiating polynomial sums, products, and compositions](/en/mko/mko-derivative-rules/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) Four differentiation rules retain differentiability premises; this pulse evaluates them exactly for finite integer-coefficient polynomials. mko-derivative-rules · Analysis Knowledge object ## [Polynomial antiderivatives and the choice of constant](/en/mko/mko-antiderivative/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) An antiderivative differentiates to f on its declared interval; this pulse selects a zero-constant rational-coefficient representative for full-real polynomials. mko-antiderivative · Analysis Knowledge object ## [Signed polynomial definite integrals on finite intervals](/en/mko/mko-definite-integral/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) 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. mko-definite-integral · Analysis Knowledge object ## [Fundamental theorem of calculus: continuity and oriented endpoints](/en/mko/mko-fundamental-theorem-calculus/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) With continuity on all R, differentiation of the accumulating integral recovers the integrand; an eligible antiderivative gives an oriented endpoint difference. mko-fundamental-theorem-calculus · Analysis Knowledge object ## [Nonempty finite trees: connectivity, cycles, and unique paths](/en/mko/mko-tree/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) A finite simple undirected graph is a tree only when nonempty, connected, and acyclic; retain singleton, degree-one leaf, and new-edge cycle conventions. mko-tree · Combinatorics and discrete mathematics Knowledge object ## [Topological space](/en/mko/mko-topological-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) 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. mko-topological-space · Topology Knowledge object ## [Natural-number divisibility](/en/mko/mko-natural-divisibility/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) A natural number a divides b when a natural multiplier k satisfies b=a·k; the two roles of zero must be distinguished. mko-natural-divisibility · Number theory Knowledge object ## [Greatest common divisor and coprimality](/en/mko/mko-gcd-coprime/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) 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. mko-gcd-coprime · Number theory Knowledge object ## [Prime natural number](/en/mko/mko-prime-number/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) A prime natural number is at least 2 and has no positive divisors other than 1 and itself. mko-prime-number · Number theory Knowledge object ## [Natural-number congruence modulo a positive modulus](/en/mko/mko-modular-congruence/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) 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. mko-modular-congruence · Number theory Knowledge object ## [Decision problem](/en/mko/mko-decision-problem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) 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. mko-decision-problem · Mathematics of computer science Knowledge object ## [Formal statements in a proof assistant](/en/mko/mko-lean-formal-statement/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) A formal statement is a core term of type Prop obtained by successful elaboration in a fixed version, import environment and local context. mko-lean-formal-statement · Formalized mathematics Knowledge object ## [Open and closed sets](/en/mko/mko-open-closed-set/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) 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. mko-open-closed-set · Topology Knowledge object ## [Neighborhood](/en/mko/mko-neighborhood/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) A set N is a neighborhood of x exactly when N contains an open set containing x; N itself need not be open. mko-neighborhood · Topology Knowledge object ## [Continuous maps between topological spaces](/en/mko/mko-topological-continuity/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) A map is topologically continuous when every open set in its target has an open preimage in its source. mko-topological-continuity · Topology Knowledge object ## [Abstract computational model](/en/mko/mko-computational-model/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) A finite-state computational model describes possible executions through its state set, initial-state set, step relation and halt-state set. mko-computational-model · Mathematics of computer science Knowledge object ## [Algorithm correctness](/en/mko/mko-algorithm-correctness/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) Algorithm correctness separates partial correctness from termination; total correctness requires both. mko-algorithm-correctness · Mathematics of computer science Knowledge object ## [Asymptotic resource bounds](/en/mko/mko-asymptotic-resource-bound/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) Asymptotic upper, lower and tight bounds require an input-size convention, cost model and quantification over a full tail. mko-asymptotic-resource-bound · Mathematics of computer science Knowledge object ## [Proof term](/en/mko/mko-lean-proof-term/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) A proof term t has propositional type P in a fixed formal environment and context, and is checked by the kernel. mko-lean-proof-term · Formalized mathematics Knowledge object ## [Discrete-time dynamical system](/en/mko/mko-discrete-dynamical-system/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) 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. mko-discrete-dynamical-system · Dynamical systems Knowledge object ## [Ordinary differential equation](/en/mko/mko-ordinary-differential-equation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D4) An ordinary differential equation relates an unknown single-variable function and finitely many derivatives; candidate residual checks are not existence or uniqueness theorems. mko-ordinary-differential-equation · Differential equations [← Previous](/en/knowledge/?page=3#results)[1](/en/knowledge/?page=1#results) … [3](/en/knowledge/?page=3#results) 4 [5](/en/knowledge/?page=5#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=5#results) 302 results · Page 4 of 13