Sunday, 26 June 2016

calculus - Absolute function continuous implies function piecewise continuous?



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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