Are there any geometric interpretations to uniform continuity?

There are specifically the two forms, continuity and uniform continuity, I'm referring to. So a function is continuous if the graph "doesn't break," but this also applies to a uniform continuous function. What else is required for it to be uniformly continuous? I know the technical definitions (continuity: $\forall \varepsilon \, \forall x \, \exists \delta \, \forall y \, ( \, |y-x|<\delta \, \Rightarrow \, |f(y)-f(x)|<\varepsilon \, )$, uniform continuity: $\forall \varepsilon \, \exists \delta \, \forall x \, \forall y \, ( \, |y-x|<\delta \, \Rightarrow \, |f(y)-f(x)|<\varepsilon \,)$) but in geometric terms, what does it mean?

Solution 1:

There's a great theorem about uniform continuity that helped me understand it quite a bit.

If $f \colon \mathbb{R} \to \mathbb{R}$ is a function and if $\lim_{x \to \infty} f(x)/x = \pm\infty$, then $f$ is not uniformly continuous.

Basically, what this means is that the "last" uniformly continuous functions are linear. If your function grows faster than every linear function, it cannot be uniformly continuous. You can apply this intuition to similar functions as well - $f(x) = \sin(x^2)$ is not uniformly continuous because eventually its local growth outpaces an arbitrary linear function.

Edit: Here's the proof. Assume that $f$ is uniformly continuous. Then there is $\delta > 0$ such that $|x - y| < \delta \Rightarrow |f(x) - f(y)| < 1$. If $f(x)/x$ has some limit as $x \to \infty$ then $(f(x) - f(0))/x$ should have the same limit, because $f(0)/x \to 0$. Now,

$\left|\frac{f(x) - f(0)}{x}\right| = \left|\frac{f(x) - f(x - \delta) + f(x - \delta) - ... + f(\delta_0) - f(0)}{x}\right| \leq \left|\frac{f(x) - f(x - \delta)}{x} \right| + \left| \frac{f(x - \delta) - f(x - 2\delta)}{x} \right| + ... + \left| \frac{f(\delta_0) - f(0)}{x}\right| < \frac{x}{\delta}\cdot\frac{1}{x} = \frac{1}{\delta}$

so the limit cannot be infinite. I used $\delta_0$ to be some positive number less than $\delta$; it could be that $x$ is not an exact multiple of $\delta$ and this takes the remainder. In the first equality I used a telescoping sum; in the second I used the triangle inequality; in the third I just counted the number of terms.

Solution 2:

Continuity of $f$ at $a$ means that I can make $f(x)$ as close as I want to $f(a)$ by making $x$ close to $a$. Let us formalize: No matter how small an error tolerance (for all $\epsilon > 0$), one can make $f(x)$ close to $f(a)$ within this error tolerance ($|f(x)-f(a)|< \epsilon$) by finding a number $\delta$ such that if $x$ is less than $\delta$ off from $a$ then $f(x)$ is less than $\epsilon$ off from $f(a)$.

For uniform continuity, first take an example. Let our error tolerance ($\epsilon$) be $1/2$. If $f(x)$ = x, then if I want to make $f(x)$ at most $1/2$ off from $f(a)$, all I need to do is make $x$ $1/2$ off from $a$ (let $\delta = \epsilon = 1/2$). Usually, however, $\delta$ depends on $x$: if $f(x) = x^2$, then near zero all of the $f(x)$'s are fairly close to each other ($f(x)$ isn't changing very fast), so $\delta$ can be pretty big - if x is kinda close to zero, then $f(x)$ within 1/2 of zero. However, if x is near 100, $f(x)$ is changing very quickly, and thus for $f(x)$ to be close to $f(100) = 10000$, $x$ has to be really close to 100 for $f(x)$ to at most 1/2 off from $f(100)$, or $\delta$ must be teeny. Uniform continuity just says that a minimum $\delta$ works: if $x, y$ are in your interval, no matter how fast $f$ is changing, you are guaranteed that if $x$ and $y$ are within $\delta$ of each other, $f(x)$ and $f(y)$ are within $\epsilon$ of each other.