Let $f: [0, 1] \to \mathbb R$ be a continuous function. Define the local Hölder exponent $H(f, x)$ of $f$ at $x \in [0, 1]$ by

$$H(f, x) := \sup\left\{0 \leq \alpha \leq 1\mid\lim_{\delta \to 0_+} \sup_{y, z \in B_\delta (x)} \frac{|f(y) - f(z)|}{|y-z|^\alpha}\ \, \text{is finite}\right\}.$$

Let $F: \mathbb R \to \mathbb R$ be an another continuous function. We define the local Hölder exponents for $F$ in the same way. Is it true that for all $x \in [0, 1]$, we have

$$H(F \circ f, x) \leq 1 + H(f, x) - H(F, f(x))?$$