Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 64

(a) Prove that the function $F(x) = | x |$ is continuous on $(-\infty, \infty)$
(b) Prove that if $f$ is a continuous function on an interval, then so is $|f|$.
(c) If $|f|$ is continuous, does it follow that $f$ is continuous as well? I so, prove it. If not, find an example.


(a) Based from the definition of absolute value,

$F(x) = |x| = \left\{
\begin{array}{c}
x & \text{ if } & x \geq 0 \\
-x & \text{ if } & x < 0
\end{array}
\right.
$

According to the definition if the function is continuous everywhere, its left and right hand limits should be equal. So,

$
\begin{equation}
\begin{aligned}
\lim\limits_{x \to 0^+} x & = \lim\limits_{x \to 0^-} -(x)\\
0 & = -(0)\\
0 & = 0
\end{aligned}
\end{equation}
$


It shows that $\lim \limits_{x \to a^+} f(x) = \lim \limits_{x \to a^-} f(x)$, therefore $F(x)$ is continuous on $(-\infty, \infty)$


(b) If $f$ is continuous on an interval then $|f| = f$ for $f > 0$ and $|f| = -f$ for $f < 0$. Also, $|f| = 0$ if $f = 0$.
Therefore, if $f$ is a continuous function, $|f|$ is continuous as well.


(c) Suppose that $f(x) = \left\{
\begin{array}{c}
1 & \text{ for } & x \geq 0 \\
-1 & \text{ for }& x < 0
\end{array}
\right.
$

$|f|$ is continuous everywhere such that if we plugin any positive or negative numbers inside the absolute value, we get
a positive value. However, $f$ is not continuous at $x = 0$ because of jump discontinuity.