Let $(a_k)_{k \geq 0} \subset \Bbb R$ be any sequence.
Let $f : \Bbb R \rightarrow \Bbb R$ be the continuous function defined by
$$f(x) = \begin{cases} a_0 \text{ if } x \leq 0 \\ a_k + (a_{k+1}-a_k)(x-k) \text{ if } x \in [k, k+1] \end{cases}$$
i.e. $f$ is a continuous, piecewise linear function.
Assume that $f$ has a uniformly bounded derivative on $\Bbb R \setminus \Bbb N$ (i.e. the sequence $(a_{k+1}-a_k)_{k \geq 0}$ is bounded). Can we conclude that $f$ is uniformly continuous ?
If not, what if we further assume some regularity for $(a_k)_{k \geq 0}$, e.g. that $(a_k)_{k\geq 0}$ is a decreasing sequence converging to $0$ ?
Hint: verify that $f(x)=f(0)+\int_0^{x}f'(t)\, dt$ and conclude that $f$ is actually a Lipschtz function.