Suppose I have a function [itex]f(x) \in C_0^\infty(\mathbb R)[/itex], the real-valued, infinitely differentiable functions with compact support. Here are a few questions:(adsbygoogle = window.adsbygoogle || []).push({});

(1) The function [itex]f[/itex] is trivially uniformly continuous on its support, but is it necessarily uniformly continuous on [itex]\mathbb R[/itex]?

(2) I want

[tex]

\left\| f(x-t) - f \left(\frac{x}{1+\epsilon} - t \right) \right\|_\infty \to 0

[/tex]

as [itex]\epsilon \to 0[/itex]. Is that so? It seems so, intuitively, if [itex]f[/itex] is uniformly continuous.

(3) Also, is it true that

[tex]

\left\| f(x-t) - f \left(\frac{x}{1+\epsilon} - t \right) \right\|_\infty = \left\| f(x) - f \left(\frac{x}{1+\epsilon} \right) \right\|_\infty

[/tex]

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# Uniform continuity and the sup norm

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