# Continuous Function proof

• Feb 21st 2010, 03:51 PM
redsoxnation
Continuous Function proof
Prove that the function f defined by f(x)=x if x is rational and f(x)=-x if x is irrational is continuous at 0 only.
• Feb 21st 2010, 04:01 PM
Plato
Quote:

Originally Posted by redsoxnation
Prove that the function f defined by f(x)=x if x is rational and f(x)=-x if x is irrational is continuous at 0 only.

Because this is not a homework service, you need to show us some effort on your part.
We will happily give you help on what you have already done.
But do not expect us to do your work for you.
• Feb 21st 2010, 05:00 PM
patrick
Quote:

Originally Posted by redsoxnation
Prove that the function f defined by f(x)=x if x is rational and f(x)=-x if x is irrational is continuous at 0 only.

Hint: If f is indeed continuous, then f^{-1}( (0,1) ) is a non-empty open subset of R. It also contains 1/2, for instance. Since 1/2 lies in an open subset of R, look at an irrational number in a neighborhood of 1/2 and show that it doesn't map into (0,1) as originally assumed.
• Feb 21st 2010, 06:22 PM
Drexel28
Quote:

Originally Posted by redsoxnation
Prove that the function f defined by f(x)=x if x is rational and f(x)=-x if x is irrational is continuous at 0 only.

I DO NOT want to do your HW for you...Plato is right, show us some work. But, a hint that may be easier than patrick is to consider that $\displaystyle f$ is continuous implies that $\displaystyle x_n\to x\implies f(x_n)\to f(x)$