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**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?

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?