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The 60 pages that link to Continuous function, each with the reason it gives.

• DerivativeRelated: Differentiability at a point implies continuity there, but continuity alone does not guarantee a derivative.

• Uniform convergenceRelated: Uniform limits of continuous functions remain continuous, unlike arbitrary pointwise limits.

• TopologyRelated: Continuity formalizes the deformations that preserve topological properties.

• Compact spaceRelated: Continuous maps preserve compactness of the image.

• Open setRelated: This preimage condition makes openness the test for continuity.

• Topological spaceRelated: This definition makes continuity depend only on the open-set structures of domain and codomain.

• Brouwer fixed-point theoremNarrower topic: Continuity prevents the map from evading a fixed point through a discontinuous jump.

• Pointwise convergenceRelated: Pointwise limits of continuous functions need not remain continuous.

• Fixed-point theoremRelated: Continuity is a central hypothesis in topological fixed-point theorems.

• 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.

• HomeomorphismRelated: Continuity is one of the two conditions a homeomorphism must satisfy.

• Bisection methodNarrower topic: Continuity is the condition that makes an endpoint sign change certify a root.

• Algebraic topologyRelated: Continuous maps induce the algebraic comparisons used to test whether spaces have the same shape.

• IntegralRelated: A continuous function on a closed interval is Riemann integrable.

• Hausdorff spaceRelated: A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

• Measurable functionCompared with: Continuity uses topology, whereas measurability uses specified sigma-algebras.

• Lebesgue differentiation theoremRelated: For continuous functions, shrinking ball averages converge directly by continuity.

• Analog signalNarrower topic: Continuity captures the defining mathematical character of analog variation.

• Connected spaceRelated: Continuous images preserve connectedness, making continuity a key tool for proving spaces connected.

• Differential calculusRelated: Differentiability implies continuity, though continuity alone does not guarantee a derivative.

• Monotonic functionCompared with: Continuity does not ensure monotonicity; a continuous function may repeatedly rise and fall.

• Bounded functionCompared with: Continuity alone does not ensure boundedness on noncompact domains.

• HomotopyRelated: Each fixed parameter value in a homotopy gives a continuous function.

• Piecewise functionRelated: At a boundary, matching one-sided limits determine whether adjacent pieces join continuously.

• Product topologyRelated: Continuity of all coordinate projections determines the product topology.

• Riemann sumRelated: Continuous functions on closed bounded intervals are Riemann integrable.

• Stone–Čech compactificationRelated: Extensions of continuous functions characterize the points and topology of βX.

• AntiderivativeRelated: Every continuous function on an interval has an antiderivative, though it may lack an elementary formula.

• Integral calculusRelated: Continuity is a common sufficient condition for a function to be integrable.

• L. E. J. BrouwerRelated: Continuity is the key condition in Brouwer’s fixed-point theorem.

• Subspace topologyRelated: Inherited open sets determine continuity for maps defined on a subspace.

• Weierstrass approximation theoremNarrower topic: Continuity is the sole regularity assumption in the theorem.

• Convergent sequenceRelated: Continuous functions send convergent sequences to sequences converging to the image of the limit.

• Descriptive set theoryRelated: Continuous maps preserve much of the structure that descriptive classifications exploit.

• Monotone functionCompared with: Continuity concerns limits, not preservation of order.

• Darboux's theoremCompared with: A derivative can satisfy Darboux's conclusion while still failing continuity.

• MorphismRelated: Continuous functions are the morphisms in the category of topological spaces.

• Normal spaceRelated: Normality supports constructing continuous functions with controlled values on closed sets.

• Sierpiński spaceRelated: Maps into Sierpiński space correspond to open subsets of their domain.

• Secant methodRelated: Continuity near a root helps explain why nearby approximations can remain informative.

• Support (mathematics)Related: Continuity makes the nonzero set open, clarifying how its closure forms support.

• Schauder fixed-point theoremRelated: Continuity ensures approximate fixed points yield a genuine fixed point.

• Separation axiomRelated: Some separation properties are characterized by the continuous real-valued functions a space admits.

• 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.