Continuity & Differentiability (Advanced)

This session builds on the foundational concepts of limits, continuity, and differentiability, focusing on advanced problem types that integrate multiple calculus concepts. While basic continuity and differentiability are covered in earlier sessions, JEE Main increasingly tests nuanced scenarios involving piecewise functions, composite functions, and the interplay between continuity, differentiability, and integrability.

Advanced Continuity Concepts:

A function f is continuous at x = a if lim(x→a) f(x) = f(a). For piecewise-defined functions, we need LHL = RHL = f(a). The key advanced scenarios include:

1. Continuity of composite functions: If f is continuous at a and g is continuous at f(a), then g(f(x)) is continuous at a. However, f(g(x)) may not be continuous if g has a discontinuity that maps into a problematic region.

2. Continuity involving [x] and {x}: The greatest integer function [x] is discontinuous at all integers. So f(x) = [g(x)] is discontinuous wherever g(x) takes integer values. The fractional part {x} = x - [x] is discontinuous at integers with a jump from 1 to 0.

3. Continuity of |f(x)|: If f is continuous, |f| is continuous. But the converse is false — |f| can be continuous while f is not (e.g., f(x) = 1 for x rational, -1 for x irrational: |f| = 1 everywhere but f is nowhere continuous).

Advanced Differentiability Concepts:

f is differentiable at x = a if lim(h→0) [f(a+h) - f(a)]/h exists (and is finite). The left derivative f'(a-) and right derivative f'(a+) must be equal.

1. Differentiability implies continuity, but not vice versa. Classic examples: |x| is continuous but not differentiable at 0. x^($\frac{2}{3}$) is continuous but not differentiable at 0.

2. Differentiability of |f(x)|: |f(x)| is differentiable at x = a if either (i) f(a) != 0 (and f is differentiable), or (ii) f(a) = 0 and f'(a) = 0. If f(a) = 0 and f'(a) != 0, then |f(x)| is NOT differentiable at a.

3. Differentiability of max/min functions: max(f,g) = (f+g)/2 + |f-g|/2 and min(f,g) = (f+g)/2 - |f-g|/2. Non-differentiability occurs where f = g and f' != g'.

4. Higher-order differentiability: f''(a) exists only if f'(x) is differentiable at a. Piecewise-defined functions often have continuous f' but non-existent f''.

5. Rolle's and LMVT connections: If f is continuous on [a,b] and differentiable on (a,b), then Rolle's (f(a)=f(b)) guarantees f'(c)=0 for some c, and LMVT guarantees f'(c) = [f(b)-f(a)]/(b-a).

Key Tests and Criteria:

For piecewise functions at the junction point x = a:

• Continuity: check LHL = RHL = f(a)
• Differentiability: check left derivative = right derivative
• Continuous differentiability ($C^{1}$): check f' is also continuous at a

The key problem-solving concept is systematically checking continuity first, then differentiability, using one-sided limits and derivatives at junction points.