continuity of |f| and its f

Question

I want to know wether if |f| is continuous at every point of R, then f is continuous also or not?

and what about its converse? if f is continuous in every point of R, does that means |f| also continuous?

Thank you

Answer 1

If $|f|$ is continuous, $f$ can be really discontinuous. Imagine a function that is $-1$ on the rationals and $1$ on the irrationals for example.

On the other hand, if $f$ is continuous, then so is $|f|$. The proof is simple. The max of two continuous functions is continuous. And both $f$ and $-f$ are continuous. and $|f|= \max(f,-f)$.

Answer 2

For your first question: Suppose we know $f(x)$ is either 1 or -1, for every $x$. Then:

• Do we know if $f$ is continuous?

• Do we know if $\vert f\vert$ is continuous?

For your second question, given a continuous $f$, is there a nice way to break the graph of $\vert f\vert$ into continuous pieces which are connected to each other? (HINT: look at the set $\{x: f(x)=0\}$ . . .)

Answer 3

Your first question has been sufficiently covered (as has the second, really), but for the second, note that if $f(x)$ and $g(x) = \lvert x \rvert$ are continuous, then $\lvert f \rvert = g \circ f$ is a composition of continuous functions.