1. ## continuous function

I'm pretty sure this statement is false.
Let f be a defined function. If abs(f) is continuous at a, then f is also continuous at a.

Is this an accetable counterexample?
f(x) =
x + 1 if x does not equal zero
1 if x equals zero.

2. The function you have proposed is continuous everywhere.
But you do have the right idea. Try this.
$f(x) = \left\{ {\begin{array}{rr}
{ - 1} & {x < 0} \\
1 & {0 \le x} \\
\end{array}} \right.$

3. Yeah I meant
x + 1 if x does not equal zero
-1 if x equals zero.

4. Well yes. That will work also.