Concept preview

What you will be able to do

1. 01Define continuity on an open interval and on a closed interval, including the one-sided continuity required at endpoints.
2. 02Classify standard function families — polynomials, rational, root, exponential, and piecewise — by where they are continuous and where they break.
3. 03Decide whether a function is continuous on a given closed interval and justify the answer by locating every point of discontinuity.

In the previous lesson you tested continuity one point at a time. Real problems, however, ask about whole stretches of the number line: is this function continuous on $[0, 5]$? Is temperature a continuous function of time over an entire day? To answer, we promote continuity from a single-point property to an interval property. The idea is simple and powerful — a function is continuous on an interval precisely when it is continuous at every point of that interval — but the endpoints of a closed interval need special care, and knowing the continuity of standard function families lets you answer most questions almost instantly.