# Continuity and Limits Proofs

• Oct 4th 2010, 07:30 PM
mscbuck
Continuity and Limits Proofs
Question: Give an example of a function g such that g is continuous nowhere, but |g| is continuous everywhere?

Fairly simple set-up. I'm thinking that the best way to attack this would be with a piece-wise function (similar to the classic 1/q if rational, 0 if irrational), but I'm having trouble thinking of a function whose absolute value WOULD be continuous. I've done a couple of these problems with properties before, but I just am really stumped on this one. Any help is appreciated!

Thanks again,

Mike
• Oct 4th 2010, 07:42 PM
Jhevon
Quote:

Originally Posted by mscbuck
Question: Give an example of a function g such that g is continuous nowhere, but |g| is continuous everywhere?

Fairly simple set-up. I'm thinking that the best way to attack this would be with a piece-wise function (similar to the classic 1/q if rational, 0 if irrational), but I'm having trouble thinking of a function whose absolute value WOULD be continuous. I've done a couple of these problems with properties before, but I just am really stumped on this one. Any help is appreciated!

Thanks again,

Mike

Hmmm, well, the most notable thing about absolute values is that they make negative numbers positive. So going off your idea, how about g(x) = 1 if x is rational and g(x) = -1 if x is irrational?