Answer
Please see below.
Work Step by Step
As we see the graph of $f(x)=\begin{cases}x-2, & x \ge 3 \\ x-3, & x<3\end{cases}$, the function has the following left- and right-handed limits at the point $x=3$:$$\lim_{x \to 3^+}f(x)=1, \qquad \lim_{x \to 3^-}f(x)=0$$According to the definition of “continuity” of a function, a necessary condition for a function to be continuous at a point is that the limit of the function does exist at the point. But, the limit of the function $f(x)$ does not exist at the point $x=3$, by Theorem 1.10, since the left- and right-handed limits are different. Thus, the function $f(x)$ is not continuous at the point $x=3$.
