I have a 'function' $f:\mathbb{R} \longrightarrow \mathbb{R}$ defined by a few axioms, and I'd like to prove its existence as a function. From the axioms I have shown that it is continuous. I can also construct the values of $f$ on a set that is dense in its domain (i.e. in $\mathbb{R}$) . Is this enough to show that $f$ truly is a function, or have I used a circular argument?
Edit - In response to a comment here are the particulars:
The specific function (actually two functions) must satisfy the following:
Let $p>0$ be a real number and let
$ S: \mathbb{R} \longrightarrow \mathbb{R}$ and $ C: \mathbb{R} \longrightarrow \mathbb{R}$
such that
- $C(x)C(y) + S(x)S(y) = C(x-y)$
- $ S(p) = 1$
- $S(x) \geq 0$ for $x \in [0,p]$
These are meant to be the sine and cosine functions. The 'axioms' are taken from a paper by Gerson B. Robinson published in the Mathematics Magazine.
If you know $f(q)$ for all $q\in \mathbb{Q}$ where $\mathbb{Q}$ is a dense set and you know that it is continuous, you can complete the definition by defining it for all real numbers through the continuity. Like $f(x) = \lim f(q_n)$ where $q_n$ is any sequence in the dense set that converges to x. If all is well, you should be able to use your continuity results to establish that this is well-defined. The explicit values you have for $f(q)$ on $\mathbb{Q}$ must also be consistent with continuity.