existence of right/left hand derivative of convex function in the interior.

Question

Let $I$ be an intervall and $f:I\to \mathbb{R}$ convex. $\Phi_1f(x,y):= \frac{f(x)-f(y)}{x-y}$ is the difference quotient of $f$. $\Phi_1f(x,y)$ is increasing in both its variables. I know that for $x<a<y$ it follows that $\frac{f(x)-f(a)}{x-a} \leq \frac{f(x)-f(y)}{x-y}\leq \frac{f(a)-f(y)}{a-y}$, hence $\lim\limits_{y \searrow x}\Phi_1f(y,x)$ exists since its monotone and bounded from bottom, same for the left hand side limit, since it is bounded from above. My question is: I cant explain why the rigth hand and left hand limit CAN'T exist in the closed Intervall, the limits only exist for $a$ which suffices $x<a<y$, which is the interior.

Answer 1

You can take the function $$f(x) = \begin{cases} 1 & \text{if } x = 0 \\ 0 & \text{if } 0 < x < 1 \\ 1 & \text{if } x = 1\end{cases}$$ on $I = [0,1]$.