Classifying the discontinuity at x = 1

For $x \le 1$, $p(x) = x^2 - 1$, so $p(1) = 1^2 - 1 = 0$, and the left-hand limit is $\lim_{x\to 1^-} p(x) = 1^2 - 1 = 0$. For $x > 1$, $p(x) = -x+3$, so the right-hand limit is $\lim_{x\to 1^+} p(x) = -(1) + 3 = 2$.

Since the left-hand limit (0) and right-hand limit (2) both exist but are not equal, $\lim_{x\to 1} p(x)$ does not exist. Because both one-sided limits are finite but different, the graph jumps abruptly from height 0 to height 2 at $x=1$. This is a jump discontinuity at $x=1$.