an interesting function

We are given the function f:R -> R defined as follows,

f(x) = 1/n, if x is a rational number
f(x) = 0, if x is irrational

(a) first I need to find a sequence f_n of cont. functions such that f_n(x) -> f(x) as x -> infinity

(b) second I need to prove that f is continuous at each irrational point and discontinuous at each rational point.

help?

Quote:

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Originally Posted by matzerath

We are given the function f:R -> R defined as follows,

f(x) = 1/n, if x is a rational number

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Is n just some fixed constant here, or is it supposed to be some enumeration of rationals?

Do you mean that f(m/n) = 1/n where m/n is a rational in lowest terms? ie is it Thomae's function - Wikipedia, the free encyclopedia ?

oh, sorry... so if we express Q as a countable union (i.e. Q=U(r_n))
f(x) = 1/n whenever x = r_n
f(x) = 0 whenever x is irrational