Can a function be both lower (but not upper) semicontinuous and convex?

Question

Is it possible to construct such an example?

For example, can a discontinuous function $f : \mathbf{R} \rightarrow \mathbf{R}$ be also convex?

Answer 1

Convexity implies continuity except possibly at the extrema of the domain

Let $f\colon D\to\mathbb R$ be convex on $D\subseteq \mathbb R$ and let $x_0\in D$ be a point with $\inf D<x_0<\sup D$. Pick $x_L,x_R\in I$ with $x_L<x_0<x_R$. Then by convexity, for $x\in [x_L,x_0]\cap D$ we have $f(x)$ not above the line through $(x_L,f(x_L))$ and $(x_0,f(x_0))$, and also not below the line through $(x_0,f(x_0))$ and $(x_R,f(x_R))$. The corrresponding bounds hold on $[x_0,x_R]\cap D$. Being squeezed betwen two intersecting lines, $f$ must be continuous at $x_0$.

Note that you may have discontinuity at the end points of a bounded domain though.