Suppose $f$ is continuous on $[a,b]$ and $f_r'(x)=0$ for all $x\in[a,b)$. Prove that $f$ is constant on $[a,b]$.
The definition of $f_r'(x)=0$ is that $$\lim_{y\rightarrow x^+}\frac{f(y)-f(x)}{y-x} = 0$$
Now it's hard to apply mean-value theorem, since the limit is given only for one side for each point.
