The definition of uniform continuity of a real-valued function states:
A function $f\colon A\mapsto\mathbb{R}$ is uniformly continuous on $A$ iff for every $\varepsilon \gt 0$ there exists a $\delta \gt 0$ such that for every $x$ and $y$ in $A$, whenever $y \in \left(x-\delta,x+\delta\right)$, it is the case that $f\left(y\right) \in \left(f\left(x\right)-\varepsilon,f\left(x\right)+\varepsilon\right)$.
Basically how my book distinguishes this from point-wise continuity is that there exists a single $\delta$ that works for every point in the domain, so once we find that $\delta$, we know it works everywhere. On the other hand, point-wise continuity says that given a $c\in A$, there exists a $\delta$ such that the function is continuous at $c$, but all these $\delta$s may be different, perhaps depending on $c$, and we might not be able to find just one $\delta$ that works for all $c$s everywhere.
I interpret this definition a completely different way, and I want to see if my conjecture is correct. I think that functions which have bounded derivatives are uniformly continuous. That is, if the "steepness" and "shallowness" of a function is limited to a certain minimum and maximum, then the there's a sufficiently small enough $\delta$ that we can use, particularly at the steepest part of the function (say at $x_0$), such that the output stays within the $\varepsilon$-neighborhood of $f\left(x_0\right)$. Is that correct? How can I prove/disprove it?
The converse states that the derivative of a uniformly continuous function is bounded. Is that also true?
Best Answer
As Jose27 noted, uniformly continuous functions need not be differentiable even at a single point.
It is true that if $f$ is defined on an interval in $\mathbb R$ and is everywhere differentiable with bounded derivative, then $f$ is uniformly continuous. In fact, it follows from the Mean Value Theorem that such an $f$ is Lipschitz, which is much stronger.
However, if $f$ is uniformly continuous and everywhere differentiable, then $f$ need not have bounded derivative. Jose27 mentions $\sqrt x$, which would work as an example on the interval $(0,\infty)$. The function $$f\left(x\right)=\begin{cases}x^2\sin\left(\frac{1}{x^2}\right) & :x\neq 0\\ 0 &:x=0\end{cases}$$ is uniformly continuous on any bounded interval such as $(-1,1)$, but has unbounded derivative near $0$. There are also examples where $f'$ is bounded on bounded intervals, but unbounded on $\mathbb R$, while $f$ is uniformly continuous. You can show that any continuous function $f$ on $\mathbb R$ such that $\lim\limits_{|x|\to \infty}f(x)=0$ is uniformly continuous, and using this fact you can see that Nate Eldredge's example here of $\sin(x^4)/(1+x^2)$ provides such an example. Another source of examples is the question
Why if $f'$ is unbounded, then $f$ isn't uniformly continuous?