 Solution

False. Give $X=\mathbb{R}$ the discrete topology and $Y=\mathbb{R}$ its usual topology. The identity bijection $f:X\to Y$ is continuous because every inverse image is a subset of the discrete space and hence is open. But $f^{-1}:Y\to X$ is not continuous: the open set $\{0\}$ of $X$ has inverse image $\{0\}$, not open in $Y$. Thus a continuous bijection need not be a homeomorphism.