Absolute Values and Continuous Functions

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If a function f is continuous, its absolute value |f| is also continuous, but the reverse is not true. An example provided is f(x) = -1 for x < 0 and f(x) = 1 for x ≥ 0, where |f(x)| is constant and continuous, while f itself is not continuous. Another example mentioned is defining f to be -1 on all rational numbers and 1 on all irrational numbers, which also results in |f| being continuous while f is not. These examples illustrate the distinction between the continuity of a function and the continuity of its absolute value. Understanding this difference is crucial in analyzing function behavior in calculus.
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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!

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What about f(x) = -1 if x &lt; 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 :)
 

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