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I'm trying to understand the intuitive meaning of uniform continuity. According to what I understand, $ f(x) $ is uniformly continuous if it's continuous and doesn't increase too fast. If that truly is the meaning, then why is the function $ f(x) = x^2, f: [0, 100000000] \rightarrow \mathbb{R} $ uniformly continuous, it clearly increases extremely fast.

Itay Hagadi
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  • Well, how fast DOES it increase? I suspect that there's a big, fast guy out there who thinks that's not so fast. – Lee Mosher Jun 25 '22 at 22:35
  • It is fast, but it is limited. Similarly, if a function never gets bigger than 500 quadrillion, then it gets large, but nevertheless it is still bounded. – JonathanZ Jun 25 '22 at 22:36
  • The compactness of the interval stops it from increasing "too fast", it's still limited. But if we define it on $[0,\infty)$ then it will not be uniformly continuous anymore. But honestly, such examples show why intuition is not good enough, and why we need formal definitions. – Mark Jun 25 '22 at 22:36
  • Roughly speaking, continuity at a point means that, when you zoom in on the graph AT THAT POINT, the graph "comes together at that point" and any oscillations that might occur damp out as you zoom in more and more. However, the "zoom magnification" needed for some specified "closeness" of coming to together (and dampening of oscillations) might change drastically from point to point. Uniform continuity is when, for any specified "closeness" of coming together (and dampening of oscillations), there is a magnification that achieves this no matter where in the graph the magnification occurs. – Dave L. Renfro Jun 26 '22 at 07:05
  • Regarding my earlier comment, I should have said "horizontally zoom in on the graph", with the vertical scale being unchanged. And since I'm already commenting again, I may as well add that the analogue of my previous comment when zooming in equally horizontally and vertically can be used for pointwise differentiability and uniform differentiability. – Dave L. Renfro Jun 26 '22 at 09:42
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    Not marking as a duplicate, but probably of interest: What is the geometrical difference between continuity and uniform continuity? – Andrew D. Hwang Jun 26 '22 at 11:04

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Uniform continuity means that if the arguments $x$ and $y$ are close to each other then the values $f(x)$ and $f((y)$ are close to each other. It immediately suggests the formal definition: someone (who doubts the function is uniformly continuous) requires the values to be within the distance $\varepsilon =10^{-3}.$ Our task is to find a number $\delta$ such that if $|x-y|<\delta$ then $|f(x)-f(y)|<10^{-3}$ for all points $x,y.$ Assume we have managed to find $\delta.$ The doubtful person claims that we just had good luck, and he sharpens his requirement to $\varepsilon_1 =10^{-6}.$ Assume we have managed to find another, usually smaller than $\delta,$ number $\delta_1$ so that $|x-y|<\delta_1$ implies $|f(x)-f(y)|<10^{-6}.$ If we can do it for any $\varepsilon>0$ the function is indeed uniformly continuous. But if we fail for one particular value $\varepsilon>0$ the function is not uniformly continuous. For $f(x) =\sin(x^2)$ , $x\in \mathbb{R},$ we are going to fail for $\varepsilon =2,$ and consequently for positive values smaller than $2.$

We treat $\varepsilon>0$ as our oponent, while $\delta>0$ is our ally.

Concerning your example, what is large or small depends on the point of view. For some people the value $10^{16}$ can be treated as a small number (rich people). Or the units are small, like $10^{-12}$second, $10^{-12}$cent or a drop of water.The amount of $10^{16}10^{-12}=10^4$ cents is not that impressive.

Ryszard Szwarc
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