Real closed interval/Strictly increasing/Inverse function/Continuous/Fact

Let ${}I\subseteq \mathbb {R}$ be an interval and let

$f\colon I\longrightarrow \mathbb {R}$

denote a continuous, strictly increasing function.

Then the

image

${}J:=f(I)\,$

is also an interval, and the inverse function

$f^{-1}\colon J\longrightarrow I$

is also continuous.

Write a proof