Answer
The statement is false (not always true).
See counterexample below.
Work Step by Step
Piecewise functions are also specified algebraically.
For example,
$f(x)=\left\{\begin{array}{lll}
-x, & for & x\lt 0\\
& & \\
x & for & x \gt 0
\end{array}\right.$
is not defined for $x=0$, but $\displaystyle \lim_{x\rightarrow 0}f(x)$ exists, and equals 0.
