Does a Convex Function need to be Continuous I have been trying the following problem and I am very confused. If possible the problem should be solved with derivatives. If the derivative exists for all the points on the graph then it is continuous.$f'(x_{0}) = \lim_{h \rightarrow 0 } \frac{f(x_{0}+h)-f(x_{0})}{h}$ for all $  x_{0}  \in \mathbb{R}$
Question:
Let $V : \mathbb{R}\rightarrow \mathbb{R}$ be a convex function. Does it have to be continuous? Prove rigorously either way.

Hint: If $x\in\mathbb R$, then for all $\lambda \in(0,1)$,
\begin{align*}
V(x)-V(x-1)&\leq {V(x)-V(x-\lambda)\over \lambda}\\
& \leq {V(x+\lambda)-V(x-\lambda)\over 2\lambda}\\
& \leq {V(x+\lambda)-V(x)\over \lambda}\\
&\leq V(x+1)-V(x).
\end{align*}
Thus the difference quotients are bounded.
