The max of the modulus of difference of a continuous function Let $I=[a,b]$ be a closed real interval
Let $f: I \to \mathbb{C}$ be a continuous function such that $|f(x)|$ is strictly decreasing
I would like to know if is it true that
$$
\max_{x,y \in I}
|f(x)-f(y)|
=
|f(a)-f(b)|
$$

No. Set 
$$
f: [0,2] \to \mathbf{C}: x \mapsto \begin{cases}(1-x)+i \,\,& \text{ if }x \in [0,1]\\
\frac{x-1}{2}+i\,\frac{3-x}{2} & \text{ if }x \in [1,2].
\end{cases}
$$
