TITLE: Uniform continuity and derivatives
QUESTION [6 upvotes]: Is there any example of a three times differentiable and uniformly continuous function $f: \Bbb R \rightarrow \Bbb R$ such that $f^{(3)}$ is bounded but $f^{(2)}$ isn't?

REPLY [5 votes]: Here’s another answer (while I like ajr’s answer a lot): we can write, for real $t,x$, $f(x+t)-f(x)=f’(x)t+\frac{f’’(x)t^2}{2}+\frac{t^3f’’’(c_{x,t})}{6}$ for some $c_{x,t}$ between $x$ and $x+t$. As $f$ is uniformly continuous and $f^{(3)}$ is bounded, it follows that, uniformly in $x$, as $t$ goes to zero, $2f’(x)t+f’’(x)t^2$ goes to zero.
In particular, there is $\epsilon >0$ such that for every $|t| \leq \epsilon$, $2tf’+t^2f’’$ is bounded by $1$. Let $0<u<t<\epsilon$.
Then $f’’=\frac{2tf’+t^2f’’}{t(t-u)}-\frac{2uf’+u^2f’’}{u(t-u)}$ is bounded.