# If f is continuous and strictly increasing, then the function $f^{-1}:f(I)\rightarrow I$ is continuous and strictly increasing.

Let $I \subseteq \Bbb R$ be a non-degenerate open interval, and let $f:I\rightarrow \Bbb R$ be a function. Suppose that f is strictly monotone. If f is continuous and strictly increasing (or decreasing), then the function $f^{-1}:f(I)\rightarrow I$ is continuous and strictly increasing (or decreasing).

Strictly monotone means the function is either strictly increasing or strictly decreasing.
Strictly increasing means that $x<y$ implies $f(x)<f(y)$ for all $x,y\in A$.
Strictly decreasing means that $x<y$ implies $f(x)>f(y)$ for all $x,y\in A$.

Now for my attempt: