Absolute Values and Continuous Functions

[Rosey24]

Homework Statement

We recently proved that if a function, f, is continuous, it's absolute value |f| is also continuous. I know, intuitively, that the reverse is not true, but I'm unable to come up with an example showing that, |f| is continuous, b f is not. Any examples or suggestions would be appreciated. Thanks!

Homework Equations

The Attempt at a Solution

 

[CompuChip]

What about f(x) = -1 if x < 0 and 1 if x \ge 0?
Then |f(x)| = 1 for all x - about as continuous as you get them, but f is not. In fact, you could even do something pathetic like make f equal -1 on all rationals and 1 on all irrational numbers :)