KnowraContinuous functionLinked fromLinked fromThe 60 pages that link to Continuous function, each with the reason it gives.All 60Related 43Narrower topic 12Compared with 5Brouwer fixed-point theoremNarrower topic: Continuity prevents the map from evading a fixed point through a discontinuous jump.Intermediate value theoremNarrower topic: Continuity is the hypothesis that rules out skipping values.Extreme value theoremNarrower topic: Continuity ensures limiting function values are attained rather than lost at boundary points.Bisection methodNarrower topic: Continuity is the condition that makes an endpoint sign change certify a root.Analog signalNarrower topic: Continuity captures the defining mathematical character of analog variation.Weierstrass approximation theoremNarrower topic: Continuity is the sole regularity assumption in the theorem.Urysohn's lemmaNarrower topic: The lemma constructs one with prescribed values on the two closed sets.Weierstrass theoremNarrower topic: Continuity prevents gaps in function values that could block an extremum.Continuous mapping theoremNarrower topic: Continuity at the limiting value is the key condition behind the theorem.Lefschetz fixed-point theoremNarrower topic: Continuity is required for the self-map to induce maps on homology.Stone–Weierstrass theoremNarrower topic: The theorem compares an algebra of these functions with the full space of them.Tietze extension theoremNarrower topic: The theorem extends this property from a closed domain to the entire space.