Examine the following function for continuity: $f(x) = |x - 5|$.

The given function is $f(x) = |x - 5| = \begin{cases} 5 - x, & \text{if } x < 5 \\ x - 5, & \text{if } x \ge 5 \end{cases}$.
This function $f$ is defined for all real numbers.
Let $c$ be any real number. Then $c < 5$,$c = 5$,or $c > 5$.
Case $I$: $c < 5$.
Then $f(c) = 5 - c$.
$\lim_{x \to c} f(x) = \lim_{x \to c} (5 - x) = 5 - c$.
Since $\lim_{x \to c} f(x) = f(c)$,$f$ is continuous for all $c < 5$.
Case $II$: $c = 5$.
Then $f(5) = 5 - 5 = 0$.
Left-hand limit: $\lim_{x \to 5^-} f(x) = \lim_{x \to 5} (5 - x) = 5 - 5 = 0$.
Right-hand limit: $\lim_{x \to 5^+} f(x) = \lim_{x \to 5} (x - 5) = 5 - 5 = 0$.
Since $\lim_{x \to 5^-} f(x) = \lim_{x \to 5^+} f(x) = f(5)$,$f$ is continuous at $x = 5$.
Case $III$: $c > 5$.
Then $f(c) = c - 5$.
$\lim_{x \to c} f(x) = \lim_{x \to c} (x - 5) = c - 5$.
Since $\lim_{x \to c} f(x) = f(c)$,$f$ is continuous for all $c > 5$.
Conclusion: Since $f$ is continuous at all real numbers,it is a continuous function.