Redirected from Uniformly continuous function
The formal definition is as follows: a function f : M > N between metric spaces is called uniformly continuous if for every real number ε > 0 there exists a number δ > 0 such that for all x_{1}, x_{2} in M with d(x_{1}, x_{2}) < δ, we have d(f(x_{1}), f(x_{2})) < ε.
Every uniformly continuous function is continuous, but the converse is not true. Consider for instance the function f(x) = 1/x with domain the positive real numbers. This function is continuous, but not uniformly continuous, since as x approaches 0, the changes in f(x) grow beyond any bound.
If M is a compact metric space, then every continuous f : M > N is uniformly continuous.
Every Lipschitz continuous map between two metric spaces is uniformly continuous.
If (x_{n}) is a Cauchy sequence and f is a uniformly continuous function, then (f(x_{n})) is also a Cauchy sequence.
The most natural and general setting for the study of uniform continuity are the uniform spaces. A function f : X > Y between uniform space is called uniformly continuous if for every entourage V in Y there exists an entourage U in X such that for every (x_{1}, x_{2}) in U we have (f(x_{1}), f(x_{2})) in V.
In this setting, it is also true that uniformly continuous maps transform Cauchy sequences into Cauchy sequences and that continuous maps on compact uniform spaces are automatically uniformly continuous.
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