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I'm trying to understand the concept a bit more intuitively, instead of getting lost in the $\epsilon s$ and $\delta s$, and can't quite seem to grasp what kind of behavior would make a continuous function non-uniformly continuous.

When looking for examples (whether in the book or online), I usually get the same 2 easy-to-understand examples; $1/x$ and $x^2$. So yeah, loosely speaking, the slope gets steeper and steeper, forcing us to pick $\delta$ smaller and smaller.

But is this the only type of behavior that causes $f$ to not be uniform continuous? I can't really imagine what else would cause problems, i.e. is there another "behavior" that can be spotted visually, or do we have to dig into it with $\epsilon$ and $\delta$ to realize whether it is or isn't?

Chan Heni
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  • No, it's really what the exampes tell: Depending on where you look, you have to pick $\delta$ smaller and smaller (which is, by the way, precisely what the $\epsilon$-$\delta$-definition comprises) - Of course something els that would cause $f$ to not be uniformly continuos would be - i f it is not even continuos in the first place. - And another important aspect is that (necessarily) the domains of your exmaple functions are not compact. - On the other hand, unboundedness is not needed, cf. $\sin\frac1x$ – Hagen von Eitzen Feb 12 '15 at 08:11

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It's a bit more than "the slope gets steeper and steeper": $\sqrt{x}$ has unbounded slope but is uniformly continuous. The intuition really is a direct translation of the $\epsilon$ and $\delta$ statement. For $f$ to not be uniformly continuous, the graph must contain points that are arbitrarily close together in $x$ (i.e. within any $\delta$) but not close in $y$ (i.e. more than some $\epsilon$ apart).

Robert Israel
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