What constitutes a mathematical proof? My question is what constitutes a mathematical proof?
I ask because I am currently doing a Calculus course in University and I am constantly confused regarding what I'm allowed to assume within a proof. Here is an example:
"Prove that f(x) = sqrt(|x| + x^2) is continuous in Real numbers"
And the "proof" from our math book: "f is continuous, because we may present it as a union of continuous functions h(x) = |x| + x^2 and g(y) = sqrt(y)" Why are we allowed to assume that h(x) and g(y) are continuous? I know it's fairly obvious that they are, but it's also fairly obvious that f(x) is continuous. Yet we cannot simply assume that f(x) is continuous.

I've got a question for you. If we are given two functions $f$ and $g$ such that both $f$ and $g$ are continuous. Can you prove that:
1) $f+g$ or $f-g$ is continuous?
2) $f*g$ is continuous?
3) $f\circ g$ is continuous ($\circ$ is the compositions function meaning  $f\circ g = f(g(x))$ )?
4) Is $f/g$ continuous if $g(x)\neq 0$?
I ask because if you can prove these in a general case then your question becomes rather trivial, as it is a direct corollary to these. That is if they can be proven. Do you want to give that a try? 
Little warning you will need to be careful with the maps of the functions $f$ and $g$ (that is the image and preimage of the functions). However that maybe going above your understanding so far.
