Checking continuity at a point using the three-part definition

First, evaluate $z(0)$: since $0 \geq 0$, we use the rule $z(x) = x^2+1$, giving $z(0) = 0^2+1 = 1$. So condition (1) is satisfied — $z(0)$ is defined and equals $1$.

Next, find the one-sided limits. As $x \to 0^-$, $z(x) = |x+1| \to |0+1| = 1$. As $x \to 0^+$, $z(x) = x^2+1 \to 1$. Since both one-sided limits equal $1$, $\lim_{x\to 0} z(x) = 1$ exists (condition 2 satisfied).

Finally, compare the limit to the function value: $\lim_{x\to 0} z(x) = 1 = z(0)$, so condition (3) is also satisfied. All three conditions hold, so $z(x)$ is continuous at $x=0$ — there is no discontinuity to classify.

The first condition checked was whether z of zero is defined. This matches the standard order of the three-part continuity test, where you first confirm the function value exists before examining limits. In the solution, z of zero was computed directly from the rule for x greater than or equal to zero, giving a value of one. Only afterward were the one-sided limits evaluated and compared.