# Uniform continuity proof $f(x)=e^{-x^2}$

Is the function $f(x)=e^{-x^2}$ uniformly continuous on $[0,\infty)$?
I'm fairly sure that it is uniformly continuous but I'm having a lot of trouble proving it using the $\epsilon$-$\delta$ proof. Any help would be much appreciated!

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For an $\epsilon-\delta$ proof, differentiate $e^{-x^2}$ to see that is has bounded derivative on $[0,\infty)$. This will tell you how to get your $\delta$ for any given $\epsilon$.

For a more general result, think of $e^{-x^2}$ as a continuous, monotone, and bounded function on $[0,\infty)$, and see if that relates to uniform continuity.

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Well, $x \mapsto \sin (x^2)$ is bounded, but I think it is not UC. –  Siminore Apr 18 '14 at 11:04
You are right, I meant to include that the function needs to be monotone. –  user133631 Apr 18 '14 at 11:14

Try to prove the following general result: if $f \colon \mathbb{R} \to \mathbb{R}$ is continuous and $\lim_{x \to \pm\infty} f(x)=0$, then $f$ is uniformly continuous. See: this thread.

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