Classifying the discontinuity in a piecewise shipping cost function

First evaluate the one-sided limits at $w=10$. Since $C(w)=5$ for $w$ near 10 from the left (including at $w=10$), $\lim_{w\to 10^-} C(w) = 5$. Since $C(w)=8$ for $w$ slightly greater than 10, $\lim_{w\to 10^+} C(w) = 8$.

Because the left-hand limit (5) and right-hand limit (8) are both finite but unequal, $\lim_{w\to 10} C(w)$ does not exist, so $C$ fails the second continuity condition. Since both one-sided limits exist but differ, the discontinuity is a jump discontinuity at $w=10$, corresponding to the sudden price increase once the order exceeds 10 lbs.

The answer is jump discontinuity. A sudden step up in value, where the left and right limits both exist but differ, describes exactly what happens to the shipping cost at ten pounds. This contrasts with a hole, which would be removable, or an unbounded rise, which would be infinite.