Continuity
Lessons in this chapter
About Continuity
Continuity connects the limit ideas from the previous chapter to functions whose graphs have no gaps, jumps, or holes.
Introduction to Continuity defines what it means for a function to be continuous at a point, before Analytically Determining Continuity gives the precise three-part test for checking it algebraically.
Properties of Continuous Functions covers how continuity behaves under combinations of functions, like sums and products.
Intermediate Value Theorem closes the chapter with a powerful guarantee: a continuous function must take on every value between two of its outputs somewhere in between.
The four lessons, from Introduction to Continuity to Intermediate Value Theorem, each open as a practice worksheet worked in the page. Problems are scored as you answer them, including the three-part continuity check, which is the piece most often half applied. Free to practice online.
Frequently asked questions
Introduction to Continuity describes a curve that can be drawn without lifting the pen, and names the ways it can break. Analytically Determining Continuity turns that into the three-part test using the limit, the function value, and their agreement, Properties of Continuous Functions covers what stays continuous when functions are added, multiplied, or composed, and Intermediate Value Theorem guarantees that a continuous function takes every value between two of its outputs. This is Unit 1 of the AP Calculus AB course framework, Limits and Continuity, and it sits beyond what the Common Core asks for in high school.
A function is continuous at a point when its graph has no break, jump, or hole there. You could draw through that point without lifting your pencil.
A function is continuous at a point when three conditions all hold: the function is defined there, its limit exists there, and that limit equals the function’s actual value there.
Sums, differences, products, and quotients (where the denominator isn’t zero) of continuous functions are themselves continuous. This lets you build up continuity for complicated functions from simpler pieces.
The Intermediate Value Theorem guarantees a continuous function takes on every value between two of its outputs somewhere in between those two input points.
Checking only that a function is defined at a point, without also checking that its limit exists there and matches: all three parts of the continuity test must hold.