uniformly continuous

uniformly continuous (not comparable)

1. (mathematical 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 ${\displaystyle D_{X}(x,y)<\delta }$, it must be the case that ${\displaystyle D_{Y}(f(x),f(y))<\epsilon }$ (where D_X and D_Y are the metrics of X and Y, respectively).

   A uniformly continuous function is a function whose derivative is bounded.

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.

• uniform continuity