Let $f:(a,b)\rightarrow(c,d)$ be a strictly increasing and continuous function. Then $f$ has an inverse function $g:(c,d)\rightarrow(a,b)$, which is also strictly increasing and continuous, and the graph of $g$ may be obtained by interchanging the axes, or, equivalently, by reflecting the graph of $f$ in the line $y=x$.

Proof. If $(x,y)$ is a point on the graph of $f$, then $y=f(x)$. Hence $x=g(y)$, so $(y,x)=(y,g(y))$ is on the graph of $g$. The range of $f$ is $(c,d)$, so $g(y)$ is defined for all $c<y<d$. The reflected graph is also continuous.