Calculus 12 · Mathematics · Lesson 1.3
Continuity
- Level
- Grade 12
- Reading time
- 15 min reading
- Review
- Last reviewed 2026-09-03
- Standard
- Accuracy standard
- Errata
Abstract
Pin down when a graph has no breaks, so the big theorems apply.
What does it mean for a function to be continuous?
Short answer
A function is continuous at a point if it has no break there: the function is defined at the point, its limit exists, and the limit equals the function’s value. Informally, you can draw a continuous function without lifting your pen. Breaks come from holes, jumps, or vertical asymptotes, which make the function discontinuous.
Concept preview
What you will be able to do
- 01Apply the three-part definition of continuity to test a function at a point.
- 02Classify a discontinuity as removable, jump, or infinite from the behaviour of the function.
- 03Determine the intervals on which polynomial, rational, root and piecewise functions are continuous.
- 04Calculate a constant that makes a piecewise function continuous at a boundary.
- 05Justify a conclusion using the Intermediate Value Theorem and diagnose a flawed continuity argument.
Informally, a function is continuous over an interval if you can trace its whole graph without lifting your pen from the paper. There are no holes to hop over, no sudden jumps between heights, and no places where the curve races off to infinity. This picture is the right intuition to carry, but intuition alone is not precise enough to build calculus on, because a function can look smooth and still hide a single missing point that no eye would catch from a rough sketch. The whole business of this lesson is to replace the informal pen-and-paper idea with an exact test, one built directly from the limits you have already studied, and then to see what that exact notion of continuity buys us in the theorems that follow.
School Plan
Continuity
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Related topics
Frequently asked questions
- What are the conditions for continuity at a point?
- The function is defined there, its limit exists there, and the limit equals the function value.
- What kinds of discontinuities are there?
- Removable (a hole), jump (a sudden step), and infinite (a vertical asymptote).