Unit 2 of 10

Continuity

  • 6 lessons
  • Parts 27 to 38

A function is continuous when its graph has no breaks, jumps or holes on the stretch you care about. Continuity is what lets you trust that a function takes every value in between, and many big theorems in calculus quietly depend on it.

Here you will define continuity precisely, sort discontinuities into types, see how continuous functions combine, and meet the intermediate value theorem and uniform continuity.

Every lesson is a pair of videos. The theory part explains the idea and the practice part works through problems. Each lesson page has the definitions and an example for every key term.

Continuity of a Function

  1. Continuous Functions

    What it means for a function to be continuous at a point, at an endpoint of its domain and on a whole interval, stated with limits and checked on concrete functions.

    Theory part 27Practice part 28

  2. Discontinuities and Continuous Extensions

    How to sort the ways a function can fail to be continuous into removable, jump, infinite and oscillating types, and how to fill a removable gap with a continuous extension.

    Theory part 29Practice part 30

  3. Graphs of Continuous Functions

    How to read continuity, one-sided continuity and each type of discontinuity off a graph with open and closed dots, and why a plotted graph alone can never prove continuity.

    Theory part 31Practice part 32

Continuity Theorems

  1. Continuity Theorems

    Why sums, products, quotients and compositions of continuous functions stay continuous, and why a continuous function on a closed bounded interval reaches a largest and a smallest value.

    Theory part 33Practice part 34

  2. The Intermediate Value Theorem

    Why a function that is continuous on a closed interval takes every value between its endpoint values, how that proves equations have solutions, and what goes wrong without continuity.

    Theory part 35Practice part 36

  3. Uniform Continuity

    The difference between continuity at each point and uniform continuity on a set, why continuous functions on closed bounded intervals are uniformly continuous, and why that matters for integrals.

    Theory part 37Practice part 38