Q.A wholesaler's pricing policy is: for an order of $x$ units, the total cost is $C(x) = 50x$ rupees if $x \leq 100$, and $C(x) = 45x$ rupees if $x > 100$ (the lower rate of ₹45 per unit applies to the entire order once it exceeds 100 units, not just the units beyond 100).

A quantity-discount policy that drops the price on the entire order once a threshold is crossed can create a genuine jump in the total-cost function at that threshold, worth checking before assuming cost functions are always smooth.

(i) Continuity at $x=100$.

$C(100)$: since $x \leq 100$ uses the first piece, $C(100) = 50(100) = ₹5{,}000$.

Left-hand limit (using $50x$ for $x \leq 100$, approaching from below): $\lim_{x \to 100^{-}} C(x) = 50(100) = ₹5{,}000$

Right-hand limit (using $45x$ for $x>100$, approaching from above): $\lim_{x \to 100^{+}} C(x) = 45(100) = ₹4{,}500$

Since LHL ($₹5{,}000$) $\neq$ RHL ($₹4{,}500$), $\lim_{x \to 100} C(x)$ does not exist, so $C(x)$ is discontinuous at $x=100$ — specifically, a jump discontinuity, since the two one-sided limits genuinely disagree (unlike Exercise 5's removable case).

(ii) Comparing 100 units with 101 units.

At $x=100$: $C(100) = 50 \times 100 = ₹5{,}000$.

At $x=101$: since $101>100$, the discounted rate applies to the whole order: $C(101) = 45 \times 101 = ₹4{,}545$. …

Students sometimes assume every real-world cost function must be continuous because 'cost rises smoothly with quantity' — a genuine quantity-discount cliff (discount applying to the whole order, not just the marginal units beyond the threshold) is a real and co …