So I have the following problem. Let $f:[0,1] \rightarrow [-1,1]$ such that $\lvert f(x) - f(y) \rvert \leq \lvert x - y \rvert$ for all $x,y \in [0,1]$. An example of such a function is drawn in Figure 1.
Now I am interested in the maximal distance between function $f$ and a piecewise linear function $h$. I will focus on the interval $[a,b] \subset [0,1]$ where $h$ is simply a linear approximation of $f$. Now I want to know what the maximal distance between function $f$ and the linear line is (see Figure 2). I know that this maximal distance is located at $c$, where the tangent line of the function $f$ is equal to the linear line.
Now the smoothness constraint on the top should give me some information on how large the difference, $\ell$, can be. However, I have not been able to find a tight bound. I think it should be something like, \begin{align} \sup_{x\in[0,1]}\lvert f(x) - h(x)\rvert < \frac{1}{2(b-a)} \end{align}
Any help is greatly appreciated. Please let me know if something about the problem is unclear.