On the existence of a non-constant sequence whose differentiable image converges

Question

Let $f: [a,b] \to \mathbb R$ be a function differentiable in $(a,b)$ , then is it true that there is a non-constant

sequence $(x_n)$ in $(a,b)$ such that the sequence $\big(f(x_n)\big)$ is convergent ?

Answer 1

If $f$ is differentiable, then it is continuous: we have $$f'(x) = \lim_{h \to 0 }\frac{f(x+h)-f(x)}h$$so in particular, $f(x+h) \to f(x)$ as $h \to 0$.

So if $(x_n)$ is any convergent sequence in $(a,b)$ with $x_n \to x$, then $f(x_n) \to f(x)$.