Math 2.0

General topology Proposition

Epsilon-delta uniform continuity

Builds on Metric spaceConvergence of a sequence in a metric space

Statement

Let (X,dX)(X, d_X) and (Y,dY)(Y, d_Y) be metric spaces and let f ⁣:X→Yf \colon X \to Y be a function. Then the following two conditions are equivalent:

  1. For each real ε>0\varepsilon > 0 there is a real δ>0\delta > 0 such that dY(f(x),f(y))<εd_Y(f(x), f(y)) < \varepsilon for all x,y∈Xx, y \in X with dX(x,y)<δd_X(x, y) < \delta.
  2. For all sequences (xn)(x_n) and (yn)(y_n) in XX with dX(xn,yn)→0d_X(x_n, y_n) \to 0, also dY(f(xn),f(yn))→0d_Y(f(x_n), f(y_n)) \to 0.

The map ff is uniformly continuous when it satisfies these conditions.

Remarks

Condition 1 is the epsilon-delta definition of uniform continuity. Here δ\delta depends only on ε\varepsilon, not on the points. Each uniformly continuous map is continuous. The converse fails: x↦x2x \mapsto x^2 on ℝ\mathbb{R} is continuous but not uniformly continuous. Take xn=n+1/nx_n = n + 1/n and yn=ny_n = n; then xn2−yn2>2x_n^2 - y_n^2 > 2. On a compact space, each continuous map to a metric space is uniformly continuous: this is the Heine–Cantor theorem. Each Lipschitz map is uniformly continuous.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Uniform_continuity, Definition of uniform continuity
  2. Web https://en.wikipedia.org/wiki/Uniform_continuity, Other characterizations
About this page
Tag0FHSDCQACY
URIm2://0FHSDCQACY
Kindstatement, proposition
WritingW0 stub
MSC54E40
StatusThis copy of the site has no build data. Run m2 status 0FHSDCQACY.
UsesMetric space, Convergence of a sequence in a metric space

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