If $f(x)$ and $g(x)$ are continuous at $x=c$ then show that: $f(g(x))$ and $g(f(x))$ are also continuous at $x=c$.

Question

If $f(x)$ and $g(x)$ are continuous at $x=c$

then show that:

$h(x)=f(g(x))$ is also continuous at $x=c$. (Given that $c$ belongs to the Domain of $h$)

Answer 1

Another counter example: let $f(x)=\frac{1}{x+1}$ and $g(x)=x-1$. Both are continuous at $x=0$, but $f \circ g$ is not. In order for your statement to be always true, then we need both functions to be continuous on $(-\infty, \infty)$ or for $f(c)=c$ and $g(c)=c$.

Answer 2

Let $g:\mathbb R\to\mathbb R$ be the function $x\mapsto x+1$ and let $f:\mathbb R\to\mathbb R$ be prescribed by $x\to x$ if $x<1$ and $x\mapsto x+1$ otherwise.

Then $g$ is continuous at every $0\in\mathbb R$ but can the same be said about $f\circ g$?