Given $f: A \rightarrow \mathbb{R}$, show that
$A=[-4,-1)\cup[0,6],$
$f(x)= \begin{cases}x+1 & \text { if } x \in[-4,-1)\text {, } \\ x & \text { if } x \in[0,6] ;\end{cases}$
is injective and find the image of $A$ under $f$. Is $f^{-1}$ continuous?
I know I’m missing something very obvious. I was able to show injectivity and $f(A)$, but is there a way to directly answer whether $f^{-1}$ is continuous based on that work?
Please provide a hint or key theorem, not the work itself.