Jumpstart - Questions and Answers Week 3

Question 1 (James Shade, incoming Fall 2018)

I am having a hard time coming up with a good intuitive understanding of what uniform continuity represents. I understand the definition and how it differs from the definition of standard continuity, but I am having a hard time identifying is the bigger picture of the information uniform continuity captures. I do notice a relationship between uniform continuity of a differentiable function and its derivative. My thinking is this: if the function's derivative is unbounded, then for any $\delta$, we should be able to choose $x$ and $y$ such that $d(x,y) < \delta$, but $d(f(x),f(y))$ as far apart as we desire. Hence, the function cannot be uniformly continuous. Is my thinking on this correct? Can anyone help me determine the right way to conceptualize uniform continuity?

Followup question (Matthew Hirning, incoming Fall 2018)
I was also thinking about uniform continuity in terms of a bounded derivative, and the nexample of $x^2$ makes perfect sense. I was wondering if anyone could explain conceptually why the unbounded derivative in the case of $\sqrt{x}$ is not an issue. I'm trying to envision it and am having trouble.

What you are saying is actually true if the function were linear: if the slope becomes infinite, the function looses continuity. For general functions, though, the slope of the tangent at a point can get large without the function loosing continuity. Second what happens at a point can happen anywhere. There are pretty bad functions, like, continuous everywhere but differentiable nowhere (think of a path of Brownian motion, which is Hölder continuous but nowhere differentiable).
As for uniform continuity, think of the fact that continuous functions are uniformly continuous on compact sets. Thus continuous, but not uniformly so, functions have to be looked for on non-compact sets. In finite dimensions, this means on non-closed or non-bounded sets.

Comment (Michael Hehman, incoming Fall 2018)
I'm not sure that it is true that if a function has unbounded derivative, then it is not uniformly continuous. For example, $f(x) = \sqrt{x}$ is uniformly continuous on $(0, \infty)$, despite the fact that $f'(x)$ is unbounded on $(0,\infty)$. There is a relationship between uniform continuity and the derivative being bounded, but it goes the other way: if a function has bounded derivative on an interval, then it is uniformly continuous on that set. I would also be curious as to how others conceptualize uniform continuity though. I also tend to think about it in terms of derivatives. For example, $x^2$ is not uniformly continuous since as $|x|$ gets large, $x^2$ becomes arbitrarily steep (derivative is unbounded), and so no $\delta$ can be found that would work for all $x \in \mathbb{R}$ at once. However, the $\sqrt{x}$ example shows that unbounded derivative is not always sufficient to determine uniform continuity.

Comment (Wes Whiting, incoming Fall 2018)
Uniform continuity is useful because regular continuity isn't strong enough to guarantee certain nice conclusions; for example, you might hope that a continuous function would send Cauchy sequences to Cauchy sequences, but this is not the case (consider the function $\frac{1}{x}$ and the sequence $\frac{1}{n}$). A uniformly continuous function does preserve the Cauchy property . Every function with a bounded derivative is Lipschitz by the mean value theorem, and Lipschitz functions are uniformly continuous. However, not all uniformly continuous functions are Lipschitz; Michael gives a good counterexample above. A wider class of uniformly continuous functions is those which are Hölder-continuous, but this is still not all of them. Intuitively speaking, a differentiable function is uniformly continuous as long as the derivative never gets too large for too long, so that while the derivative may be very large at a given point, it drops off quickly enough that you can still bound the change in $y$ for a sufficiently small change in $x$. This line of thought, while in principle correct, can be misleading since it is not precise enough, especially the use of the expression "too large for too long".

Differentiability is way more than continuity. To make a parallel: if you look for people who are 5ft or taller, you can also look for people who are 6ft or taller and you would be fine. You would be missing very many people, though. Differentiable functions with bounded derivative are way more special than just continuous: there are continuous functions that are nowhere differentiable. A finite slope can give you an indication of how continuous a function is at the point, but continuity at a point does not imply existence of a tangent slope. The function $$ f(x)=\begin{cases} x^\alpha\sin(1/x),&x>0,\\ 0,& x=0,\end{cases} $$ is continuous in $x=0$ for any $\alpha>0$ but check out the derivative (even for $\alpha=1$).