Past exam of the options I course of the University of Oxford / 2012 / E / E1 / a / ii

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- Solution ii

Solution

The map is uniformly continuous when, for every

ϵ > 0

, there is a

δ > 0

such that

d_{X} (x, x^{'}) < δ ⟹ d_{Y} (f (x), f (x^{'})) < ϵ for all x, x^{'} \in X .

The choice of

δ

may depend on

ϵ

but must be independent of the locations of

x

and

x^{'}

.

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