Are all functions that have an inverse bijective functions? To have an inverse, a function must be injective i.e one-one.
Now, I believe the function must be surjective i.e. onto, to have an inverse, since if it is not surjective, the function's inverse's domain will have some elements left out which are not mapped to any element in the range of the function's inverse. 
So is it true that all functions that have an inverse must be bijective? 
Thank you.

Yes. Think about the definition of a continuous mapping.
Let $f:X\to Y$ be a function between two spaces. Topologically, a continuous mapping of $f$ is if $f^{-1}(G)$ is open in $X$ whenever $G$ is open in $Y$. In basic terms, this means that if you have $f:X\to Y$ to be continuous, then $f^{-1}:Y\to X$ has to also be continuous, putting it into one-to-one correspondence. 
Thus, all functions that have an inverse must be bijective.
