The function is meant to be a periodic function with period that has zeros for every element of its domain, being the integers times the constant .
is a constant. What kind of constant? Integer, real, complex? Quaternion maybe? I'm guessing it's a real but you should not make your readers guess. Better to say:
When you say that the domain is the integer multiples of
, then that's all we know about f. As far as we know it's not defined anywhere else. But I don't think that's what you mean. Rather, I think you mean that
is defined on ALL the reals and happens to be zero on integer multiples of
. Is that right? If so:
continuous? Differentiable? What else are we assuming about
? If you have any other unspoken assumptions, they should be stated clearly.
To give a solid example, let
You should write
on the left. Otherwise we don't know what the independent variable is. Is it
? Is it
? Or maybe it's
is the independent variable, we write
I didn't look at the rest but hopefully you see what needs to be clarified. The main thing is that the domain is ALL the reals, and the zeros happen to be integer multiples of
, assuming that's what you mean. And as a general principle, try to eliminate any ambiguity in the mind of the reader.
ps -- You have a subtle error in your logic. You have
and later you ask us to consider
does not happen to be uniquely defined. So you have to either tell us what
is, or show that your argument is independent of the choice of a particular one-sided inverse of
Maybe I better say a few words about a common point of confusion. The word inverse has two different meanings when it's applied to functions.
If a function
is invertible, that means (by definition) that there exists some specific function that uniquely inverts it. (I'm omitting the technical details here). But
is not invertible. Rather for each point in the range, there are either one or two points in the domain that get mapped to that point in the range; and we can create "a", not "the" inverse by making a choice from each of the inverse images of points in the range. But this will only be a one-sided or "partial" inverse.
So you can't just use the notation
because it's not uniquely defined.
Here's more than you ever want to know about it, but you should give this page a look:https://en.wikipedia...nverse_function
Pay particular attention to the section on partial inverses.https://en.wikipedia...artial_inverses
Edited by wtf, 20 March 2016 - 02:11 AM.