I'm a little confused in my understanding of continuous topological functions. In the link below(I can't have it embedded since I'm new) the bottom left picture is what I am trying to understand. Is that technically a continuous function by transforming a line into an open circle. I understand that a line could technically be a part of a circle of radius infinity, but would forming this circle create the inside genus? Or am I horribly misunderstanding this?
On
Is that a line, or a line segment? Treating a line as a circle of infinite radius would mean that the pre-image of one of the points on the circle would be +- infinity, which is are not valid points in traditional geometry. A line can be taken to a circle continuously by taking e^ix, but that is not injective.
Technically the diagram itself is not a valid definition for a function. There are continuous functions $\Bbb R\to S^1$, and there are discontinuous functions $\Bbb R\to S^1$ as well.
Assuming that you are just wrapping the line around the circle infinitely many times, your function is e.g. $x\mapsto e^{ix}$ and is continuous (although not injective).
Assuming that you are wrapping the line once, but missing one point, again this is a continuous function, e.g. $x\mapsto e^{2i\tan^{-1}(x)}$. There, you are missing one point ($-1$), so the mapping is not surjective.
It is impossible for a mapping $\Bbb R\to S^1$ to be all three at once continuous, injective, and surjective.
Note: the formulas given are just examples, since all you have is a picture, a lot of topologically equivalent formulas would be possible in each of the above situations.