There are a few foundational results that let you do talk about derived topologies and the like.
Are you using epsilon-delta continuity, open set preimage continuity, or neighborhood continuity? They are all equivalent (on R and subsets), but they might require different technical things to deal with.
Ie, if f:A->B where A and B are topological spaces, showing that if X is a subset of A, then f|X
: X->B is continuous on the derived topology might take a technical result.
Similarly, if Image(f) is a subset of B, which is a subset of C, showing that f:A->C is continuous if f:A->B is continuous would be another technical result. (I originally wrote "iff", which I think is false -- let the image of f be a set of separate points in C. Then all sets in Im(f) are open. Which requires that the inverse image of each point in Im(f) be open, which would mean that f is continuous... ok, that doesn't disprove it. Thinking again: The derived topology of B from C results in more subsets of B being open -- so the inverse image requirement is more strict, so "only if" is going to be true. Why is "if" true? Is "if" true? It should be -- it would be nasty if it isn't. The open sets in B are exactly the open sets in C intersect B. The inverse image of an open set in C is equivalent to the inverse image of the open set restricted to B, as Im(f) is a subset of B, and inverse image is a pointwise concept whose "kernel" in a sense is Im(f)c
. So "if" also works. Ok, it is an "iff", which is nicer than the alternative.)
Make that: Similarly, Image(f) is a subset of C, which is a subset of B, showing that f:A->C is continuous iff f:A->B is continuous would be another technical result.
With those two, showing that h:= 1/2 (f + |f|) : R->R is pointwise equal to f+
, and image(h) is a subset of R+
, lets you say that h's continuity implies f+
However, I do not think that that level of rigor is all that common in introductory analysis courses, and isn't necessary beyond introductory analysis courses because the proof is merely a technical issue. This isn't a bad thing -- at some point, you really got to stop it with the rigor, or you end up with spending 100 pages building a foundation so you can define what 1+1 is, and why it equals 2 (which is an interesting result and all, but it probably doesn't belong in an intro to analysis course!) And it could easily be the case that you have proven this result already.