A function f is differentiable at x = a if lim(h->0) [f(a+h) - f(a)]/h exists (and is finite). This requires the left derivative and right derivative to be equal.

Cases where f'(a) does not exist:

1. Corner/cusp: f is continuous but left derivative != right derivative. Example: |x| at x = 0 (left derivative = -1, right derivative = 1).
2. Vertical tangent: Derivative is infinite. Example: f(x) = x^$\frac{1}{3}$ at x = 0. f'(0) = lim h^$\frac{1/3}{h}$ = lim h^(-2/3) = infinity.
3. Discontinuity: If f is discontinuous at a, it cannot be differentiable there. Continuity is necessary (but not sufficient) for differentiability.

Key theorems:

• Differentiable at a => Continuous at a (converse is false)
• f(x) = |g(x)| is non-differentiable where g(x) = 0 and g'(x) != 0

JEE favorite: f(x) = max(g(x), h(x)) or f(x) = min(g(x), h(x)). These have corners where g(x) = h(x).

For piecewise functions: Check differentiability at junction points by computing left derivative and right derivative separately.