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I'm looking for a geometric way to identify functions that are continuous but not uniformly continuous without using the definition.

I can't really put my hands on a concrete difference between the two.

Arctic Char
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Bebars Hossny
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  • Although it's sufficient, but not necessary, look for bounded derivative. For example, $f(x)=1/x$ and $x^2$ are not uniformly continuous. – Ted Shifrin Jul 18 '20 at 03:15

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If you do something like Robinson's non-standard analysis, allowing infinitely large and infinitely small numbers, you can say $x\mapsto e^x$ fails to be uniformly continuous because when $x$ is infinitely large, an infinitely small increase in $x$ can result in an increase in $e^x$ that is not infinitely small. For example, when $e^x$ increases by $1,$ since the derivative is infinite at that point, $x$ will increase by an infinitely small amount.

Likewise when $x\ne0$ is infinitely small, then $\sin\dfrac 1 x$ can go all the way from $+1$ to $-1$ with an infinitely small change in $x,$ so this function of $x$ is not uniformly continuous.

If $f(x)$ changes by an infinitely small amount whenever $x$ changes from a standard real number to a number differing from that by an infinitely small amounnt, then $f$ is continuous. If in addition, this works not only when $x$ is a standard real number but also whenever $x$ differs from a standard real number by an infinitesimal and also whenever $x$ is an infinitely large non-standard real number, then $f$ is uniformly continous.

Michael Hardy
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