$f:\mathbb{R}\to \mathbb{R}$ continuous and $\lim_{h \to 0^{+}} \frac{f(x+2h)-f(x+h)}{h}=0$ $\implies f=$ constant. Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function with the property that
$$\lim_{h \to 0^{+}} \dfrac{f(x+2h)-f(x+h)}{h}=0$$ for all $x \in \mathbb{R}$. Prove that $f$ is constant.

Hint: for a given $h$ one has
$${f(x + h) - f(x) \over h} = {f(x + h) - f(x + h/2) \over h} + {f(x + h/2) - f(x + h/4) \over h} + ....$$
$$=  {1 \over 2}{f(x + h) - f(x + h/2) \over h/2} + {1 \over 4}{f(x + h/2) - f(x + h/4) \over h/4 } + ....$$
You actually need continuity of $f(x)$ already for the above. Now take limits as $h$ goes to zero in the above carefully and conclude ${\displaystyle \lim_{h \rightarrow 0^+} {f(x + h) - f(x) \over h}} $ is always exists and is equal to zero. Then use this to show $f$ is constant.
