Answer
The statement is not always true (for piecewise defined functions with a discontinuity, for example).
Changing "specified algebraically" with "a closed function" would produce a correct statement.
Work Step by Step
Piecewise functions are also specified algebraically.
For example,
$f(x)=\left\{\begin{array}{lll}
-1, & for & x\lt 2\\
+1 & for & x \geq 2
\end{array}\right.$
is defined on all of $\mathbb{R}$, but the one-sided limits at $x=2$ differ, which means that no limit exists there.
Thus, $f$ is discontinuous at x=2.
The statement would be correct if it was about closed functions. They are continuous at every point on their domains.
