Continuity of the composition of continuous functions

Question

I have a question about the continuity of composite functions. Let $f:[a,b] \rightarrow \mathbb{R}$ and $g:[c,d] \rightarrow [0,1]$ be continuous functions. Define the function $h:[a,b]\times [c,d] \rightarrow \mathbb{R}$ as $$ h(x,y) = f\left(x\cdot g(y)\right).$$ Is $h(x,y)$ a continuous function? If it is, how to prove it?

Thanks in advance!

Answer 1

Hint: h ist the composition of three continuous functions, namely $x \mapsto f(x)$, $(x, y) \mapsto xy$ and $(x, y) \mapsto (x, g(y))$.