Absolute Values and Continuous Functions

Rosey24

1. The problem statement, all variables and given/known data

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!

2. Relevant equations

3. The attempt at a solution

CompuChip

What about [itex]f(x) = -1[/itex] if [itex]x < 0[/itex] and 1 if [itex]x \ge 0[/itex]?
Then [itex]|f(x)| = 1[/itex] 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 :)