prove that a composite function is continuous

Question

For these functions:

$$h(x)= \begin{cases} e^{-x} & x\geq 0 \\ \sqrt{|x|} & x<0 \\ \end{cases}$$

$$g(x)=(x−4)(x+1)^{2}$$

So obviously h(x) is discontinuous at x =0, but how do I prove that h(g(x)) is continuous? By the continuity definition you cannot prove h(g(x)) is continuous (since h(x) is discontinuous itself). I think a good place to start is to prove that g(x) is continuous, it is trivial how to do that. But what do I do next? I'm stuck.

Answer 1

As the product of continuous functions, $g(x)$ is continuous everywhere.

As you indicated, $h(x)$ is only discontinuous at $x=0$.

Thus $h[g(x)]$ is only discontinuous when $g(x) = 0$.

Thus $h[g(x)]$ is only discontinuous when $x = 4$ or $x = -1$.