Expert: Paul Klarreich - 11/20/2010

Question
Dirichlet's function: Let f:(0,1)->R be given by

f(x)={1/q if x=p/q in lowest terms with p,q in N(natural#s)
    {0 if x is irrational}
Prove in one line that f(x) is continuous at every irrational number but discontinuous at every rational number.  

Answer
Questioner:   brittany
Country:  United States
Category:  Calculus
Private:  No
 
Subject:  Dirichlet's Function/continuity
Question:  Dirichlet's function: Let f:(0,1)->R be given by

f(x)={1/q if x=p/q in lowest terms with p,q in N(natural#s)
   {0 if x is irrational}
Prove in one line that f(x) is continuous at every irrational number but discontinuous at every rational number.
..........................................................
Choose some small number epsilon, for which I shall write 'e', and then if x is irrational, then any neighborhood of x will contain an infinite number of rationals, because the rationals are dense, and then some one of those will have to be written p/q where q > 1/e and so f(p/q) < e, showing that lim[x0->x] f(x0) = f(x), which is the definition of continuity, but when x is rational, then f(x = p/q) = 1/q and we can take e < 1/q and find, in any neighborhood of x, some irrational number x0, since they are also dense, whose f(x0) = 0 is further away from f(p/q) than e.

Whew!  You did say one line, didn't you?

Next time, just ask for the proof, OK?