1.12 Continuity over an Interval

Continuous on an interval

A function is continuous on an interval if it is continuous at every point in it. Polynomials are continuous everywhere; rational, root, and trig functions are continuous on their domains.

Piecewise functions

For a piecewise function, continuity is most in question at the boundary points where the rule changes. Check that the pieces meet: same limit from both sides and same value there.

Key terms

Continuity on an interval

Continuous at every point of the interval.

Polynomials

Continuous everywhere (all real numbers).

Rational/root/trig

Continuous on their domains.

Piecewise check

Verify the pieces meet at boundary points.

Meeting condition

Equal one-sided limits and equal value at the boundary.

Try an AP-style question

Which function is continuous for all real numbers?

1. f(x) = 1/(x² − 1)
2. f(x) = √x
3. f(x) = tan x
4. f(x) = eˣ + x³