Continuous function.

Let f be that function defined by setting:

f(x) = x if x is irrational
= p sin(1/q) if x = p/q in lowest terms.

At what point is f continuous?

Continuous for irrational x, and for x = 0. Sketch:
p*sin(1/q) - p / q
= p(sin(1/q) -1/q)
But sin x - x = o(x^2) when x -> 0
So, for large q,
|p(sin(1/q) - 1/q)| < p (1/q)^2 = (p/q) / q

Is this correct?

id say it presents a problem since the sum of two irrational numbers has not been explicitly proven to be an irrational number, so there are numbers between every other number that have yet to be defined, and since many of those potentially rational numbers involves $\pi$ (like constant multiples of eulers constant) it might bring the possibility of creating a locally continuous point in the original space. anyhow, ill shut up since im probably totally wrong.