Q.A wholesaler's total billed cost for $x$ units of a commodity is $C(x)=100x$ for $0<x\le50$, and $C(x)=90x$ for $x>50$ (the lower rate applies to the WHOLE order once it crosses 50 units). Examine whether $C(x)$ is continuous at $x=50$, and comment on the business consequence.

The pricing rule changes abruptly at $x=50$ and applies to the entire order rather than only the units above the threshold, so the cost function should be checked for a jump at that point. …

Step 1 — Left-hand limit (approaching 50 from below, using the $100x$ branch).

Step 2 — Right-hand limit (approaching 50 from above, using the $90x$ branch).

Step 3 — Apply the continuity test. LHL ($5{,}000$) $\neq$ RHL ($4{,}500$), so $\displaystyle\lim_{x\to50}C(x)$ does not exist — $C(x)$ is discontinuous at $x=50$ (a jump discontinuity, exactly the pattern flagged in Section 5).

Dual-check — compute the actual billed cost just below and just above 50 units.

This confirms the jump numerically: cost actually FALLS from Rs. 5,000 (at 50 units) to Rs. 4,590 (at 51 units), even though one more unit was bought. …

Assuming a rate cut at a threshold is automatically "good for the buyer and harmless to the firm" without checking continuity is the real-world mistake this question is built to catch — an unexa …