Finding all continuous and discontinuous points of composite functions

Question

Let $f(x) = \operatorname{sgn}(x)$ and $g(x) = 1 + x^2$.

How do I go about finding all the continuous and discontinuous points of the functions $f\circ g$ and $g\circ f$ ?

Answer 1

Composition of continuous functions is still continuous, hence $f \circ g$ can only be discontinuous when $g=0$, which is impossible. Hence $f\circ g$ is continuous everywhere. For $g \circ f$, we need to check $x=0$, when $x$ near $0$, $g\circ f(x)=2, g\circ f(0)=1$, hence $\lim_{x\to0}g\circ f(x)=2\neq g\circ f(0)$, hence it is only discontinuous at $x=0$.