# more continuity of functions

• Apr 8th 2009, 11:38 PM
noles2188
more continuity of functions
1) Let f: D-->R and define |f|: D-->R by |f|(x)=|f(x)|. Suppose that f is continuous at c elements of D. Prove that |f| is continuous at c.

2) If |f| is continuous at c, does it follow that f is continuous at c? Justify your answer.

----I don't know where to start for 1) but for 2), I would think that f is continuous at c since it is part of |f|. Is this proper logic?

Thanks.
• Apr 9th 2009, 12:26 AM
vsywod
if f is continuous at c then $\displaystyle \forall \varepsilon >0 \exists \delta>0$ with $\displaystyle \vert x - c \vert < \delta \Rightarrow \vert f(x) - f(c) \vert <\varepsilon$.
Using the same $\displaystyle \delta$, we have
$\displaystyle \vert \vert f(x)\vert - \vert f(c)\vert \vert\leq\vert f(x)-f(c)\vert <\varepsilon$, so $\displaystyle \vert f\vert$ is continuous at c.

For 2) take a function which is -1 on $\displaystyle \mathbb{Q}$ and 1 on $\displaystyle \mathbb{R}-\mathbb{Q}$. Then $\displaystyle \vert f \vert$ is 1 and continous, but f certainly isn't.