Continuity and differentiability

The essentials

A function f is continuous at x = c if f(c) is defined, the limit of f(x) as x → c exists, and the two are equal. In practice, for a piecewise function you check that the left-hand limit, the right-hand limit and f(c) are all the same number. A function is continuous on an interval if it is continuous at every point of it. Polynomials, sin x, cos x, eˣ and |x| are continuous everywhere; rational functions are continuous wherever the denominator is not zero.

A function is differentiable at c if the limit of [f(c + h) − f(c)]/h as h → 0 exists. For a piecewise function this means the left-hand derivative equals the right-hand derivative. Differentiability implies continuity, but not the other way round: |x| is continuous at 0 yet has a sharp corner there, with left derivative −1 and right derivative +1.

Key points

• Sum, difference, product and quotient (non-zero denominator) of continuous functions are continuous.
• Chain rule: if y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx). Example: d/dx sin(3x) = 3 cos(3x).
• Graph test: a break or jump means not continuous; a sharp corner or vertical tangent means not differentiable.
• |x − a| is not differentiable at x = a.
• The greatest integer function [x] is discontinuous at every integer.

Exam tips

• For "find k so that f is continuous", equate the two one-sided limits and solve.
• For "find a and b so that f is differentiable", use two equations: one from continuity and one from equal one-sided derivatives.

Common mistakes

• Assuming a continuous function must be differentiable.
• Forgetting the inner derivative in the chain rule, for example writing cos(x²) instead of 2x cos(x²) as the derivative of sin(x²).