Continuity at a point

A function is continuous at a point if you can draw its graph through that point without lifting your pen: no gap, no jump, no hole.

To check it with numbers, find three things at x = c:

• LHL (left-hand limit): what f(x) approaches as x comes from the left. Put x = c – h and let h → 0.
• RHL (right-hand limit): what f(x) approaches from the right. Put x = c + h and let h → 0.
• f(c): the actual value at c.

f is continuous at c when LHL = RHL = f(c). All three must exist and be equal. If any one is missing or different, f is discontinuous at c.

Example: f(x) = x² for x ≤ 1, and 2x – 1 for x > 1. LHL = 1² = 1, RHL = 2(1) – 1 = 1, f(1) = 1. All equal, so continuous at 1.

Example: f(x) = x + 1 for x ≤ 2, and 2x for x > 2. LHL: f(2 – h) = 3 – h → 3. RHL: f(2 + h) = 4 + 2h → 4. They differ, so the graph jumps at 2: discontinuous.

For a piecewise function, each piece on its own is usually a polynomial, which is fine. Only the joining points need checking.

Common facts: polynomials, constants and |x| are continuous everywhere. The greatest integer function [x] jumps at every integer.

Finding an unknown (k, a, b) so that f is continuous: write LHL = RHL = f(c) and solve. Two unknowns need two such conditions, usually from two joining points.