Discuss the continuity of the function defined by $f(x) = \begin{cases} x + 2, & \text{if } x < 0 \\ -x + 2, & \text{if } x > 0 \end{cases}$

(N/A) Observe that the function is defined at all real numbers except at $x = 0$. The domain of definition of this function is $D = D_1 \cup D_2$,where $D_1 = \{x \in \mathbb{R} : x < 0\}$ and $D_2 = \{x \in \mathbb{R} : x > 0\}$.
Case $1$: If $c \in D_1$,then $\lim_{x \to c} f(x) = \lim_{x \to c} (x + 2) = c + 2 = f(c)$. Hence,$f$ is continuous in $D_1$.
Case $2$: If $c \in D_2$,then $\lim_{x \to c} f(x) = \lim_{x \to c} (-x + 2) = -c + 2 = f(c)$. Hence,$f$ is continuous in $D_2$.
Since $f$ is continuous at all points in its domain,we conclude that $f$ is continuous on its domain. Note that the function is not defined at $x = 0$,so we do not discuss continuity at $x = 0$.