Answer
False.
Work Step by Step
If the absolute value of a function is continuous, it does not necessarily mean the function is continuous.
Suppose that
$f(x) = 2$ when $x \geq 0$
$f(x)=-2$ when $x<0$
$|f(x)|$ = 2 everywhere, so it is continuous everywhere. But for $f(x)$, there is a jump discontinuity at $x=0$, so $f(x)$ is not continuous everywhere.
