# Continuety of splitted function

1. Dec 29, 2008

### transgalactic

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

Last edited by a moderator: May 3, 2017
2. Dec 29, 2008

### HallsofIvy

Staff Emeritus
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.

3. Dec 30, 2008

### transgalactic

i know the definition of bound and of continuity
i cant understand what the last hint means and how to use it
??