Homeomorphism definition I was told by my professor that homeomorphisms are continuous maps with continuous inverse, but do those conditions also imply that the map is bijective?
It was inconvenient that your professor worded it this way. From the get-go, saying that a homeomorphism is continuous function with a continuous inverse assumes that we have some function $f$, and that its inverse, $f^{-1}$, exists. Off-the-bat, just cause we have a function $f$, it does not mean that it has an inverse. 

In addition, the way your professor phrased it doesn't make it clear whether we are talking about a subset of the range, or the whole range itself: in a homeomorphism, we must have the whole range, which is something your professors phrasing neglected to capture. 

If I were you, I would just forget about what your professor said. It is kind of circular. Just to make it clear. By definition, a homeomorphism is a function $h$, from a topological space $X$, to a topological space $Y$, such that the following hold: 


$h$ is 1-1 
$h$ is onto
$h$ is continuous
$h^{-1}$ is continuous


Note, we do not say $h^{-1}$ exists here: this is a consequence of $h$ being 1-1, as we can always create $h^{-1}$ in such a case. ALSO, we imply that the image of $h$ is all of its range: this is captured by saying that $h$ is a bijection.
A function that has an inverse is injective, or one-to-one.  Every one-to-one function is onto some set.  So a continuous function with a continuous inverse is a homeomorphism onto some set even if there is some larger target set.

However, if $f:A\to B$ is continuous and has an inverse and its inverse is also continuous, that does not mean $f$ is a homeomorphism from $A$ to $B$.  Rather, $f$ is a homeomorphism from $A$ to some subset of $B$, which may or may not be all of $B$.

Among the simplest instructive examples of a continuous bijection whose inverse is not continuous is $\theta\mapsto e^{i\theta}$ from $[0,2\pi)$ to the circle.