Continuity and Differentiation Notes for Class 12 & IIT JEE

Continuity means a function has no breaks, holes, or jumps at a point — the limit exists and equals the function value. Differentiability means the function has a well-defined tangent (slope) at that point. Every differentiable function is continuous, but a continuous function like |x| may fail to be differentiable at sharp corners.

According to NTA's official JEE Main paper structure, Continuity and Differentiability typically accounts for 2–3 questions (8–12 marks) in every session. It is one of the most consistently tested topics in the Mathematics section, making it a high-priority chapter for any student targeting above 85 percentile.

Yes, |x| is continuous at x = 0. The LHL = 0, RHL = 0, and f(0) = 0 — all three match, so the continuity condition is satisfied. However, it is not differentiable at x = 0 because the left-hand derivative is −1 and the right-hand derivative is +1, and they are not equal.

Logarithmic differentiation means taking the natural log of both sides of y = f(x) before differentiating. Use it whenever the function has the form [f(x)]^(g(x)) — a variable base raised to a variable exponent. Examples include y = xˢⁱⁿˣ or y = (tan x)^(ln x). Applying the standard chain rule directly to such functions is not possible.

Both theorems appear at the end of NCERT Chapter 5. Rolle's Theorem states that if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then at least one c exists in (a, b) where f'(c) = 0. The Mean Value Theorem generalises this: f'(c) = [f(b) − f(a)]/(b − a) for some c in (a, b). Both are frequently tested in JEE Main.