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.

• What you may use depends entirely on the context of the course. There is not really a mathematical answer to this.
– Eff
Mar 31, 2017 at 22:34
• You are right - the book should really prove that $g$ and $h$ are proper. Perhaps it does, in an earlier section? (To elaborate on @Eff's comment: while there is indeed a precise notion of mathematical proof, the question of what your class considers an "acceptable proof", or what is accepted as a proof by the mathematical community in general, does not admit a precise answer.) Mar 31, 2017 at 22:34
• That's a very interesting question! Normally when you're in a class that requires an axiomatic approach, they start with certain assumed concepts and move from there. However, this isn't the case with some calculus courses. In general, don't assume it; prove it. Well unless you've proven something before. Then you can refer to your earlier proof. Mar 31, 2017 at 22:38
• @NoahSchweber Yep, totally true. Just to clarify upon my comment: I mean that often when we do math in practice, assumptions of already known truths exist without them being explicitly stated. I think the easiest thing to do is simply ask the teacher what you may assume.
– Eff
Mar 31, 2017 at 22:38
• The proof depends on the knowledge (theorems) you have acquired previously and the theoretic context. Mar 31, 2017 at 23:40

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.