## uniformly-continuous

Adjective

(*not comparable*)

- (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 , it must be the case that (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.*

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*.