real analysis - Continuity and its inverse function

Let $f: [0,1) \cup [2,3] \to \mathbb R$

$f(x) =
\begin{cases}
x &: x\in [0,1)\\
x-1&: x\in[2,3]
\end{cases}$

Show that $f$ is continuous and strictly increasing in $[0,1) \cup [2,3]$ and that its inverse function $f^-1 : [0,2] \to [0,1) \cup [2,3]$ is discontinuous in $x_0 = 1$

My thoughts:

I think that I can show the continuity by one-sided limits, and strictly increasing by $x