Додаток: рівномірна неперервність

Continuity at a point (which you may or may not have seen yet) is defined as follows:

A function $f$ is continuous at a point $p$ if for every $\epsilon > 0$, there exists a $\delta > 0$ so that every $x$ within $\delta$ of $p$ is within $\epsilon$ of $f(p)$.

However, continuity at a particular point $p$ is a "local" condition, in that we only care what's happening "near" $p$. But for any particular $\epsilon$, the $\delta$'s can vary drastically among different particular values of $p$. (Indeed, we may write $\delta = \delta(\epsilon,p)$ to emphasize that $\delta$ may depend on $\epsilon$ and $p$.) For example, let's consider the function $f(x) = x^2$.

Let's take $\epsilon = 1$.

Okay! Let's try a different value of $p$.

But $\sqrt{10001}-100$ is a lot smaller than $1$. To see how much smaller, compute $$\sqrt{10001} - 100 = \frac{1}{\sqrt{10001} + 100} \approx \frac{1}{200}.$$

So even though $x^2$ is continuous everywhere, it's a lot "more continuous" or "flatter" at some points than others.

In light of this, we can define uniform continuity as follows.

A function $f$ is uniformly continuous if for EVERY $\epsilon > 0$, we can pick a $\delta$ which works for ALL points $p$.

The key change is that $\delta$ can only depend on $\epsilon$, and not $p$. So we might think of uniform continuity as a "global" condition, in contrast to the "local" condition of continuity at a particular point $p$, or the "half-local/half-global" condition of continuity (i.e. continuity at every point $p$).

This means that uniform continuity is a "stronger" condition than continuity (i.e. continuity at every point $p$). So if a function is uniformly continuous, it must be continuous. What about the other way around?

Nope, far from it -- the larger $x$ is, the more sensitive $f$ is to small changes in $x$. (See if you can prove this!)

That's uniform continuity!

(Video) Continuous and Uniformly Continuous Functions

by Steve Stein

Here's a video by Steve Stein explaining what uniform continuity is and how it differs from regular continuity.