Continuous functions between metric spaces Continuous function

for lipschitz continuous function, there double cone (shown in white) vertex can translated along graph, graph remains entirely outside cone.

the concept of continuity functions between metric spaces can strengthened in various ways limiting way δ depends on ε , c in definition above. intuitively, function f above uniformly continuous if δ not depend on point c. more precisely, required every real number ε > 0 there exists δ > 0 such every c, b ∈ x dx(b, c) < δ, have dy(f(b), f(c)) < ε. thus, uniformly continuous function continuous. converse not hold in general, holds when domain space x compact. uniformly continuous maps can defined in more general situation of uniform spaces.

a function hölder continuous exponent α (a real number) if there constant k such b , c in x, inequality

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{\displaystyle d_{y}(f(b),f(c))\leq k\cdot (d_{x}(b,c))^{\alpha }}

holds. hölder continuous function uniformly continuous. particular case α = 1 referred lipschitz continuity. is, function lipschitz continuous if there constant k such inequality

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{\displaystyle d_{y}(f(b),f(c))\leq k\cdot d_{x}(b,c)}

holds b, c in x. lipschitz condition occurs, example, in picard–lindelöf theorem concerning solutions of ordinary differential equations.