Classifying the discontinuity of a step-priced cost function at t = 2

For $t$ slightly less than 2, such as $t = 1.9$, we have $\lceil t \rceil = 2$, so $C(t) = 4(2) = 8$. This holds for all $t$ in $(1, 2)$, so $\lim_{t \to 2^-} C(t) = 8$. For $t$ slightly greater than 2, such as $t = 2.1$, we have $\lceil t \rceil = 3$, so $C(t) = 4(3) = 12$, and this holds for all $t$ in $(2, 3)$, giving $\lim_{t \to 2^+} C(t) = 12$.

Since the left-hand limit ($8$) and right-hand limit ($12$) both exist but are not equal, $\lim_{t \to 2} C(t)$ does not exist, even though both one-sided limits are finite. This is precisely the signature of a jump discontinuity: the function value 'jumps' from 8 to 12 as $t$ passes through 2, matching the visible vertical jump in the step graph at $t=2$.