Continuety of splitted function

[transgalactic]

prove that this function continues on every irrational point
and discontinues in every rational point?
http://img136.imageshack.us/img136/5985/87260996wu4.png

 

[HallsofIvy]

The English for "splitted function" is "piecewise defined" function although here it is more "pointwise" defined:
f(x)= 0 if x is irrational, 1/n if x is ration, x= m/n reduced to lowest terms (and n is assumed to be positive).

As I said in a previous response, lim_{x\rightarrow a} f(x)= L if and only if lim_{n\rightarrow\infty} f(a_n)= L for any sequence {a_n} converging to x. Obviously for any sequence, a_n., of irrational numbers converging any x, that limit is 0.

So in order that this be continuous, the limit as we approach along rational numbers must also be 0 and the function value must be 0.

Here's a hint. If x= m/n, given any \epsilon> 0 there are only a finite number of possible N such that N< 1/epsilon (so 1/N> 1/\epsilon) and for each such N there are only a finite number of M such that M/N is within \delta of m/n. Use that to prove that the limit always exists and is always equal to 0.

 

[transgalactic]

i know the definition of bound and of continuity
i can't understand what the last hint means and how to use it
??