Does a Convex Function need to be Continuous

Question

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.

Answer 1

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.