Are there any geometric interpretations to uniform continuity?

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.

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.

I hope that this helps: by an inscribed triangle I mean a right triangle s.t. the vertices of its longest side are on the graph of function $f$ and the adjacent sides to the right angle (legs) are parallel to $x$-axis and $y$-axis(unfortunately I don't know how to draw a picture here), we call the side parallel to $x$-axis its base and ,the side parallel to $y$-axis its altitude (just here not generally!)

$f$ is uniformly continuous if when we look at the set of all inscribed triangles, the altitude becomes small enough provided that the base becomes small enough.

for example $f(x)=\sin{1/x}$ is not uniformly continuous because near $0$ you can find inscribed triangles with arbitrary small base s.t. their altitude is not small enough. similarly, $f(x)=x^2$ is not uniformly continuous (I assume the domain is the whole real line) because you can find inscribed triangles with similar property, however, in this case the triangles go far away from $0$.

One can find a similar interpretation for continuity in terms of inscribed triangles which have one of their vertices (other than the vertex with the right angle) at point $(x,f(x))$.