The intuitive meaning of uniform continuity

Question

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.

Answer 1

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.