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Uniform continuity

A single input tolerance controls output error uniformly across the domain.

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Different ways to understand it

• Uniform Continuity: An Introduction

  Mike, the Mathematician · English · Explanation
  Connection evidence

  Reviewed English material v1: card "Definition of Uniform Continuity" at 0:10 — A function $f: E \to \mathbb{R}$ is uniformly continuous on $E$ if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $x, y \in E$, $|x - y| < \delta$ implies $|f(x) - f(y)| < \varepsilon$. Crucially, $\delta$ depends only on $\varepsilon$, not on the specific points $x$ and $y$.

• Functional Analysis 18 | Compact Operators

  The Bright Side of Mathematics · English · Explanation
  Connection evidence

  Reviewed English material v1: card "Uniform continuity of the kernel" at 4:30 — Because the kernel $k$ is continuous on the compact square $[0,1]\times[0,1]$, it is uniformly continuous. The board states the epsilon-delta form: for every $\varepsilon>0$ there exists $\delta>0$ such that whenever $\|(s_1,t_1)-(s_2,t_2)\|<\delta$, one has $|k(s_1,t_1)-k(s_2,t_2)|<\varepsilon$. Input distance is Euclidean in $\mathbb R^2$; output distance is absolute value in $\mathbb R$.

Knowledge connections

• Epsilon–delta limits · Related · Map context
• Continuity · Related · Map context

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