If $h$ is continuous and $h=g \circ f$, are $g$ and $f$ continuous?

Question

If $h$ is continuous and $h=g \circ f$, are $g$ and $f$ continuous?

I know if $f$ and $g$ are continuous, then $f\circ g$ is also continuous. But I have a composition of functions and I don't have information of the functions.

Answer 1

No. Define $f,g : \mathbb{R} \to \mathbb{R}$ by $g(x) = 1$ and $f(x) = \mathbf{1}_{\mathbb{Q}}(x)$ (or your favourite discontinuous function). Then $(g \circ f) (x) = 1$.