Uniform continuity

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y be sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and y themselves.
Continuous functions can fail to be uniformly continuous if they are unbounded on a finite domain, such as



f
(
x
)
=



1
x





{\displaystyle f(x)={\tfrac {1}{x}}}
on (0,1), or if their slopes become unbounded on an infinite domain, such as



f
(
x
)
=

x

2




{\displaystyle f(x)=x^{2}}
on the real line. However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map).
Although ordinary continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbourhoods of distinct points, so it requires a metric space, or more generally a uniform space.

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