AHL 5.12 (HL)—Continuity, differentiability and first principles

Continuity and differentiability are local conditions.

Continuity requires the function value and both one-sided limits to agree; differentiability additionally requires matching finite slopes.

Example

|x| is continuous at 0 but not differentiable there because its left slope is −1 and right slope is 1.

Check the function, limit and derivative separately at the suspected point.

Differentiability implies continuity, but continuity alone does not imply differentiability.

A limit converges when nearby function values approach one finite value; it diverges when no finite common value exists. For a polynomial, first principles gives $f'(x)=\lim_{h\to0}[f(x+h)-f(x)]/h$; for $f(x)=x^2$, simplifying produces $2x+h\to2x$. Higher derivatives may be written $d^ny/dx^n$ or $f^{(n)}(x)$. At a point, differentiability implies continuity, but a corner, cusp, vertical tangent or discontinuity prevents a finite two-sided derivative.