$f$ continuous in $[a,b]$ and differentiable in $(a,b)$ without lateral derivatives at $a$ and $b$ Does anyone know an example of a real function $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ such that the lateral derivatives
$$ \lim_{h \to a^{+}} \frac{f(x+h)- f(x)}{h} \quad \text{and} \quad \lim_{h \to b^{-}} \frac{f(x+h)- f(x)}{h}$$
don't exist?

A semicircle centered at the midpoint of $[a,b]$ with radius $(b-a)/2$ is an example:
$$f(x)=\sqrt{ [(b-a)/2]^2 - (x-[(a+b)/2])^2 }.$$
