The arcsine is "not really a function" My math teacher said that later on in life, if you pursue math, you would see that the arcsine function isn't actually a function. I do not see how it's not a function.
 A: Not only is $\sin$ a perfectly well-defined function in the set-theoretic sense, we actually have an effective formula to compute the values of this function. Taking a wild guess here on what the teacher might have meant, perhaps it would be that relying on the geometric meaning of the sine function is not a terribly accurate way of doing things. That would be a correct statement. There is a very rigorous way to define the sine function but it relies on a pretty heavy theorem.
Here we go: The sine function is the unique function $f\colon \mathbb R \to \mathbb R$ satisfying the initial condition $f(0)=0,f'(0)=1$ and the differential equation $f''=-f$. In order for this to be a legitimate definition we invoke a theorem from the theory of function that guarantees that this particular differential equation does have a unique solution given the initial condition.
So, the sine function exists as a particular case of a theorem that allows us to construct many other functions. Now, from this information about the sine function one can find its Taylor series, compute its radius of convergence to be $\infty$, and use the remainder form in order to compute $\sin(t)$ for any value of $t\in \mathbb R$ and to any degree of accuracy (given sufficient computational time).
I would not say that the existence of the sine function is trivial. But it certainly is a function. Perhaps it is worth noting here that the common definition of $\sin(t)$ as the $Y$ coordinate of the point on the unit circle at $t$ radians from the positive $X$ axis measuring counterclockwise is correct but it relies on the notion of distance/angle. That is a tricky business. These geometric subtleties are avoided by the theorem mentioned above, essentially converting the whole thing to a more algebraic situation and good dose of analysis.
A: Following clarification from the OP that the arcsine function had been referred to: the arcsine is "not really a function" because there are infinitely many solutions to (for example) $\sin x=0$, so $\arcsin0$ could be any one of infinitely many values, whereas a function must map one input to one output. $\arcsin$ is therefore, stricto sensu, a multifunction.
Implementations of the arcsine in programming languages must choose one principal value out of infinitely many to serve as the aforementioned $x$, which leads to branch cuts over which that principal value is not continuous. For the arcsine's case, those branch cuts most often are $(-\infty,-1)$ and $(1,\infty)$.
