Metric Space Continuity

Real Analysis · Axiom Academy

LESSON Metric Space Continuity One ε–δ definition for every space: replace absolute value with a distance, and continuity follows everywhere. 1. The ε–δ Definition, with Balls Let (X, d_X) and (Y, d_Y) be metric spaces and . The classical definition reads the same as on — only the rulers change. A function is continuous at a when every output tolerance ε can be met by some input tolerance δ: Pick any ε-ball around f(a) in Y there is a δ-ball around a whose image lands inside it The animation makes this concrete with a genuine map: the contraction on the plane, with a=(2,1) and f(a)=(2,0) . It is Lipschitz with constant , so exactly — the entire δ-ball is shrunk by half and slid onto f(a) . 2. One Definition, Many Spaces The power of the metric formulation is that a single definition specializes to every setting you have met. Each space just supplies its own d ; the ε–δ machine is untouched. d(x,y) = |x-y| . Recovers the classical ε–δ definition exactly — absolute value is a metric. . The , , balls are diamonds, circles, squares — yet on all give the same continuous functions (the norms are equivalent). d(x,y)=0 if x=y , else 1 . Take : then forces x=a , so . Every function is continuous! , or via integrals. In infinite dimensions these are not equivalent — the metric choice changes which sequences converge. Example: the integration operator

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