Key Relationship: Differentiability at a point implies continuity at that point, but NOT vice versa.

Proof: If f'(a) exists, then lim [f(a+h)-f(a)] = lim h * [f(a+h)-f(a)]/h = 0 * f'(a) = 0. So f is continuous.

Counterexamples (Continuous but NOT Differentiable):

• f(x) = |x| at x = 0 (corner)
• f(x) = x^$\frac{1}{3}$ at x = 0 (vertical tangent)
• f(x) = x^$\frac{2}{3}$ at x = 0 (cusp)
• f(x) = x*sin$\frac{1}{x}$, f(0)=0 at x = 0 (oscillating slope)

The Hierarchy: Differentiable < Continuously differentiable < Twice differentiable < ... < Smooth ($C^{infinity}$)