Continuity at a point

Informally, a function is continuous if its graph can be drawn around a point without lifting the pen. The precise test uses limits.

A real function f is continuous at a point c of its domain when lim f(x) as x → c equals f(c). Three things must hold: f(c) is defined, the limit exists, and the two are equal.

The limit exists only when the left-hand limit (x approaching c from below) and the right-hand limit (from above) are both finite and equal. So continuity at c means LHL = RHL = f(c).

Worked example: f(x) = 2x + 3 at x = 1. The limit is 2(1) + 3 = 5 and f(1) = 5, so f is continuous at 1.

Worked example: f(x) = x³ + 3 for x ≠ 0, with f(0) = 1. Both one-sided limits at 0 are 3, but the value is 1, so f is discontinuous at 0. Redefining f(0) as 3 would repair it.

Worked example: |x| at 0. From the left |x| = −x → 0, from the right |x| = x → 0, and |0| = 0, so |x| is continuous at 0 even though its graph has a corner there.

Worked example: the function equal to x + 2 for x ≤ 1 and x − 2 for x > 1. At 1 the left-hand limit is 3 and the right-hand limit is −1, so the graph jumps and f is discontinuous at 1.

Finding a constant: kx + 1 for x ≤ 5 and 3x − 5 for x > 5 is continuous at 5 only if 5k + 1 equals 15 − 5 = 10, which gives k = 9/5.