How do you prove that proof by induction is a proof? Are proofs by induction limited to cases where there is an explicit dependance on a integer, like sums? I cannot grasp the idea of induction being a proof in less explicit cases. What if you have a function that suddenly changes behavior? If a function is positive up to some limit couldn't I prove by induction that it is always positive by starting my proof on the left of the limit?
I feel like I am not getting a grasp on something fundamental and I hope you can help me with this. Thank you.

This math.SE question has a lot of great answers as far as induction over the reals goes. And as Austin mentioned, there are many cases in graph theory where you can use induction on the vertices or edges of a graph to prove a result. an example:
If every two nodes of $G$ are joined by a unique path, then $G$ is connected and
$n = e + 1$ Where $n$ is the number of nodes and $e$ is the number of edges. 
$G$ is connected since any two nodes are joined by a path. To show $n = e + 1$,
we use induction. Assume it’s true for less than $n$ points (this means we're using strong induction). Removing any edge from $G$
breaks $G$ into two components, since paths are unique. Suppose the sizes are $n_1$ and $n_2$,
with $n_1 + n_2 = n$. By the induction hypothesis, $n_1 = e_1 + 1$ and $n_2 = e_2 + 1$; but then$$n = n_1 + n_2 = (e_1 + 1) + (e_2 + 1) = (e_1 + e_2) + 2 = e − 1 + 2 = e + 1$$
So our statement is true by strong induction
