Differentiability at a point — check:

1. Is f continuous? (Necessary condition)
2. Do left and right derivatives exist and match?
3. Use the DEFINITION, not lim f'(x)

Non-differentiable points — look for:

• Corners: |x-a| terms
• Cusps: x^$\frac{2}{3}$ type at zero
• Vertical tangents: x^$\frac{1}{3}$ type at zero
• Jumps: [x], {x} at integers

Key Results:

• Differentiable => Continuous (NOT vice versa)
• f' cannot have jump discontinuities (Darboux)
• n roots of f => >=n-1 roots of f'
• LMVT: |f(a)-f(b)| <= max|f'| * |a-b|
• IVT: sign change => root exists

Parameter Finding:

• Continuity equation: match LHL, RHL, f(a)
• Differentiability equation: match left and right derivatives

Common Values:

• |x| at 0: continuous, not differentiable
• x|x| at 0: differentiable, f'=0, f'' DNE
• $x^2$ sin$\frac{1}{x}$ at 0: differentiable, f'=0, f' discontinuous