calculus - Is composition of surjective continuous function with discontinuous function discontinuous?

Let $I_1,I_2,I_3$ be intervals $\subset \mathbb{R}$. Suppose $f:I_1 \to I_2$ is a surjective continuous function and $g: I_2 \to I_3$ is a discontinuous function. Must the composition $g \circ f$ be discontinuous?

There are some easy counter-examples if $f$ is not assumed to be surjective, e.g. taking $f$ to be a constant function, or in a way that "dodges" the discontinuous point(s) of $g$.

However if such "dodging" is prohibited, I fail to construct such functions nor find an answer from many similar questions on this site. So I am interested to know whether counter-examples exist? If not, is there a proof? Does it have something to do with intermediate value theorem?