Counter example that the composition of two lower semicontinuous functions is not lower semicontinuous.

Question

Is there a simple counter example for this?

Answer 1

Let $f(x)=0$ for $x\neq0$ and $f(x)=-1$ for $x=0$. Clearly $f$ is l.s.c. Let $g(x)=-x$ which is continuous. Then $g(f(x))$ is clearly not l.s.c.

So an easy source of counterexamples is the fact that decreasing functions reverse upper and lower semicontinuity.