Students learn to differentiate long before they learn to justify it, and the board asks for the justification.
Updated 5 September 2026 · Delhi Home Tutor
Almost every Class 12 student can differentiate. Far fewer can show a function is continuous at a point, or explain why differentiability implies continuity but not the reverse. Those are the questions that separate marks in this chapter, and they are skipped because they look like theory rather than technique.
| Class | 12 |
|---|---|
| Subject | Maths |
| Skipped because | It looks like theory, not technique |
| Where marks go | Checking continuity at a point, LHD and RHD |
| Board | CBSE, ISC |
| Enquiries | +91 92121 42427 |
The three-line check nobody writes
Continuity at a point has a fixed structure: left limit, right limit, value of the function. Equal on all three, and the function is continuous. Students know this and still write a paragraph of reasoning instead of the three lines that earn the marks.
Insisting on the three-line format, every time, is the single highest-return habit in the chapter. It is fast, it is unambiguous, and it maps directly onto the marking scheme.
Differentiability, and why the order matters
Left-hand derivative and right-hand derivative, computed from the definition, compared. Students who jump straight to differentiating the formula miss the entire point of the question — at a piecewise junction there is no single formula to differentiate.
The relationship runs one way only. Differentiable implies continuous; continuous does not imply differentiable. The modulus function at zero is the standard counterexample and it is worth having ready, because questions asking for one appear regularly and a student who has to invent it on the spot loses time.
Piecewise functions with a parameter to determine are the most common question type. The method is mechanical once the student accepts it: set the pieces equal at the junction for continuity, set the one-sided derivatives equal for differentiability, solve.
What to drill
- The three-line continuity check, written out, on every question.
- LHD and RHD from first principles at a junction, not by differentiating the formula.
- Two standard counterexamples held ready — modulus at zero, and a function with a jump.
- Rolle and Lagrange stated with their conditions checked explicitly before use.
The mean value theorems in particular are lost to unchecked conditions. Marks are awarded for verifying the function is continuous on the closed interval and differentiable on the open one, and students who go straight to the conclusion forfeit them.
How much time it needs
Less than students fear and more than they give it. Three or four focused sessions covers the standard question types comfortably, provided the sessions are spent writing proofs rather than watching them.
Questions parents ask
My child can differentiate anything but scores badly here.
That is the expected pattern. Differentiation is technique; this chapter tests justification, which is a different and teachable skill.
Are the theorems really examinable?
Yes, and the condition-checking attached to them carries marks in its own right.
Is first-principles differentiation still needed?
At piecewise junctions, yes — there is no formula to apply, so the definition is the only route.
How does ISC differ?
It generally expects the written reasoning laid out more fully. The three-line habit serves both boards well.
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