Advanced Continuity and Differentiability builds on foundational concepts to address nuanced JEE problems involving piecewise functions, absolute values, composite functions, and the interplay between various calculus concepts. This topic carries 1-2 questions per year, often integrated with other calculus concepts.

The fundamental hierarchy is: Differentiability implies Continuity, but NOT vice versa. Classic counterexamples are |x| (continuous, not differentiable at 0), x^$\frac{1}{3}$ (continuous, vertical tangent at 0), and x^$\frac{2}{3}$ (continuous, cusp at 0). The function x*sin$\frac{1}{x}$ with f(0)=0 is continuous but not differentiable at 0, while $x^2$*sin$\frac{1}{x}$ with f(0)=0 IS differentiable at 0 but f' is discontinuous there.

For piecewise functions at junction point x = a: first check continuity (LHL = RHL = f(a)), then check differentiability (left derivative = right derivative). When finding parameters, these give two equations. Important: use the DEFINITION f'(a) = lim [f(a+h)-f(a)]/h, not lim f'(x) as x->a.

The absolute value function |f(x)| is not differentiable where f(x) = 0 AND f'(x) != 0 (simple zeros create corners). At double roots where f'(a) = 0, |f(x)| remains differentiable. For sum of absolute values |x-a| + |x-b| + ..., each term contributes a non-differentiable point.

Max(f,g) and min(f,g) are non-differentiable where f = g and f' != g'. Use the formula max(f,g) = $\frac{f+g+|f-g|}{2}$.

The greatest integer function [x] is discontinuous at all integers. Composite functions [g(x)] are discontinuous where g(x) crosses integer values. The fractional part {x} has period 1 with jumps at integers.

Rolle's Theorem and LMVT are the primary tools for proving existence results. If f has n roots, f' has at least n-1 roots. LMVT provides bounds: |f(a)-f(b)| <= M|a-b| where M = max|f'|. The Intermediate Value Theorem proves root existence: if f is continuous and changes sign, it has a zero.

Darboux's Theorem constrains derivatives: f' always has the intermediate value property, so it cannot have jump discontinuities. This means any discontinuity of f' must be oscillatory (like the cos$\frac{1}{x}$ term in the derivative of $x^2$ sin$\frac{1}{x}$).