Prove or give a counterexample: the composite of a continuous function and a discontinuous function is discontinuous.

Question

If $f, g : \mathbb{R} \to \mathbb{R}$ are functions with $f$ continuous and $g$ not continuous, then $g \circ f$ is not continuous.

Answer 1

$g(x)=1$ for $x\ne 0$, $g(0)=0$, $f(x)=1$, then $g\circ f(x)=1$.

Answer 2

$g(x)=0$ for $x=0$ and $g(x)=1$ everywhere else. $f(x)=1$. Then you get $g \circ f =1$ continuous.