Continuous functions and breaks

A function is called continuous when it is continuous at every point of its domain. On a closed interval [a, b], continuity at a means right-continuity and continuity at b means left-continuity.

Constant functions, the identity x and every polynomial are continuous everywhere. So is |x|.

f(x) = 1/x is continuous: it is not defined at 0, so 0 is not a point of its domain and is not tested. Near 0 its values blow up, reaching 10ⁿ when x = 10⁻ⁿ, with one-sided limits −∞ and +∞.

A point where f is not continuous is a point of discontinuity, and it must lie in the domain. The function equal to x + 2 for x < 0 and −x + 2 for x > 0 is continuous, because 0 is outside its domain.

The function equal to x for x ≥ 0 and x² for x < 0 is continuous everywhere: both pieces tend to 0 at 0 and f(0) = 0.

The greatest integer function [x] is continuous at every non-integer and discontinuous at every integer c, where the left-hand limit is c − 1 and the right-hand limit is c.

Finding two constants: 5 for x ≤ 2, ax + b between 2 and 10, and 21 for x ≥ 10 is continuous when 2a + b = 5 and 10a + b = 21, so a = 2 and b = 1.