# uniformly continuous

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## English

### Adjective

uniformly continuous ‎(not comparable)

1. (analysis, of a function from a metric space X to a metric space Y) That for every real ε > 0 there exists a real δ > 0 such that for all pairs of points x and y in X for which $D_X (x, y) < \delta$, it must be the case that $D_Y (f(x), f(y)) < \epsilon$ (where DX and DY are the metrics of X and Y, respectively).
A uniformly continuous function is a function whose derivative is bounded.

#### Usage notes

This property is, by definition, a global property of the function's domain. That is, there is no such thing as "uniform continuity at a point," since the choice of δ for a given ε does not depend on where the points x and y are located in X.