I have a simple true/false question that I am not sure on how to prove it.
If |f(x)| is continuous in ]a,b[ then f(x) is piecewise continuous in ]a,b[
Anyone that can point me in the right direction or give a counterexample, even though I think it's true. Thanks in advance!
Answer
Here is a counter example to the statement:
Define f(x) to be 1 if x is rational and −1 if x is irrational. Now f is not continuous anywhere, but |f| is identically 1 and thus continuous.
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