Question
A taxi charges ₹ for the first , then ₹ per additional km. Express the fare as a function of distance , and check whether it is continuous and differentiable at .
Solution — Step by Step
Left-hand limit: .
Right-hand limit: .
Function value: . All three are equal, so is continuous at .
Left-hand derivative: for , so LHD .
Right-hand derivative: for , so RHD .
LHD RHD, so is not differentiable at .
The fare graph has a sharp corner at — the rate of change jumps from to rupees per km. Continuous (no break in the price), but not smooth (sudden change in slope).
Final answer: is continuous at but not differentiable there.
Why This Works
Continuity asks “is there a break in the graph?” Differentiability asks “is there a corner?” Real-world piecewise pricing functions are usually continuous (otherwise you’d see ₹ for and ₹ for , which makes no business sense), but corners are common at price-tier boundaries.
This is the same reason absolute value is continuous but not differentiable at .
Alternative Method
Plot the function. A V-shape, kink, or corner means continuous but not differentiable. A jump means not continuous (and hence also not differentiable).
For CBSE 4-mark questions, always check three things at the boundary: LHL, RHL, for continuity; LHD and RHD for differentiability. Tabulate them — examiners give partial marks for the framework even if arithmetic slips.
Common Mistake
Assuming “continuous implies differentiable”. The reverse is true (differentiable implies continuous), but not this direction. The standard counterexample is at .