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If one gives you a continuous function $f(x)$ on a set, let's say the interval $[a, b]$. Are there any fast ways to intuitively see if $f(x)$ is uniformly continuous on $[a, b]$?

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    $\begingroup$ In practice, it should be relatively easy to verify that $f$ has bounded derivative, and this is sufficient. (This isn't limited to an interval.) $\endgroup$ – Qiaochu Yuan Oct 3 '12 at 4:48
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$f$ is continuous at every real $x$ if for every real $x$ and for every infinitely small $dx$, the quantity $f(x+dx)-f(x)$ is also infinitely small. But $f$ is uniformly continuous on the real line if for every infinitely small $dx$, the difference $f(x+dx)-f(x)$ is infinitely small even if $x$ is not real, i.e. $x$ could be infinitely close to a real number or $x$ could be infinitely large.

Thus, $x\mapsto\sin\frac1x$ fails to be uniformly continuous because if $x$ is infinitely close to $0$, an infinitely small change in $x$ can change $\sin\frac1x$ from $1$ to $-1$, and the difference is not infinitely small. And $x\mapsto e^x$ is not uniformly continuous because if $x$ is infinitely large, $e^x$ can increase by $1$ when $x$ increases by an infinitely small amount.

All of this is made precise in Robinson's non-standard analysis.

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If $f(x)$ is continuous and the interval $[a,b]$ is closed and bounded, then there is a theorem that says that $f(x)$ will always be uniformly continuous!

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  • $\begingroup$ In fact, for any compact domain. $\endgroup$ – Rudy the Reindeer Oct 3 '12 at 6:08

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