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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.

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

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

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

• 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.

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

• 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.

• 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.

• 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.

• 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 metrization theoremRelated: The constructed functions encode topological separation as real-valued data.

• General topologyRelated: Continuity is the central relation through which topological properties are compared.

• Ham sandwich theoremRelated: Continuity lets the balancing argument track how each side's measure changes with the cut.

• Iterated integralRelated: Continuity on suitable bounded regions provides familiar conditions for repeated integration.

• Slutsky's theoremRelated: Continuity supports the theorem’s extension to transformations beyond basic arithmetic.

• Hairy ball theoremRelated: Continuity prevents a tangent field from changing direction through a jump.

• Regula falsiRelated: Continuity makes opposite signs at interval endpoints evidence of an interior root.

• Peter–Weyl theoremRelated: Uniform density is asserted within the continuous functions on the group.

• Vector-valued functionRelated: Continuity of a vector-valued function can be checked in its components.

• Fundamental lemma of the calculus of variationsRelated: Continuity upgrades nonvanishing at one point to a neighborhood where a localized test detects it.