Let $a<b$ be real numbers and $f:(a,b)\to\mathbb{R}$ be a continuous function into $\mathbb{R}$ equipped with the standard topology. Show that the graph \begin{equation}\Gamma_f=\{(x,f(x)):x\in(a,b)\}\end{equation} is homeomorphic to $\mathbb{R}$ equipped with the standard topology.
Attempt: in order to show a function $g:(X,\tau_X)\to (Y,\tau_Y)$ is a homeomorphism, we need to show it is bijective, continuous, and that $g^{-1}$ is continuous. In this problem, we need to show that $f$ satisfies these conditions. It is continuous as given in the problem, but the remaining two axioms that it needs to satisfy aren't clear. I wonder if I've misunderstood and that it's not $f$ we needs to show satisfies the axioms, but I cannot see what else.