Class 12 Mathematics Chapter 5 Continuity and Differentiability – Revision Notes

Continuity and Differentiability builds from the basic definition of a continuous function up to the chain rule, implicit differentiation, and theorems such as Rolle’s Theorem and the Mean Value Theorem.

Last Updated: September 23, 2026

Common Mistakes Students Make in Continuity and Differentiability

Continuity

• f continuous at a: LHL=RHL=f(a).
• Differentiability ⇒ continuity (not vice versa).

Differentiation Rules

• Chain rule: [f(g(x))]’=f'(g(x))g'(x).
• Implicit differentiation: differentiate both sides w.r.t. x, solve for dy/dx.
• d/dx(e^x)=e^x; d/dx(log x)=1/x.
• Logarithmic differentiation for [f(x)]^g(x).

Theorems

• Rolle’s Theorem: f(a)=f(b) ⇒ ∃c with f'(c)=0.
• Mean Value Theorem: f'(c)=[f(b)−f(a)]/(b−a).

One-Line Summary

Continuity requires matching limits and function value; differentiability is a stronger condition implying continuity, governed by chain/implicit/logarithmic differentiation rules and mean value theorems.

Quick visual: a worked diagram from the full Extra Questions page, for reference.

Frequently Asked Questions

What does it mean for a function to be continuous at a point?
A function is continuous at a point if the limit of the function as it approaches that point exists, and that limit equals the actual value of the function at that point, meaning there are no breaks, jumps, or holes in the graph there.

Is every differentiable function continuous, and is the converse true?
Every differentiable function is continuous, but the converse is not true; a function can be continuous at a point yet not differentiable there, such as at a sharp corner like the modulus function at zero.

What to Revise First (and Last) in This Chapter

Written by Satish

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