Continuity
A function is continuous if you can draw it without lifting the pencil.
About this topic
A function is continuous at a point if you can draw it through that point without lifting your pencil: f(a) exists, the limit exists, and they are equal. Discontinuities may be removable holes, jumps or infinite asymptotes.
Polynomials, sine and cosine are continuous everywhere. The Intermediate Value Theorem says that a continuous function on [a, b] takes every value between f(a) and f(b). So if f(1) = −2 and f(2) = 3 for a continuous f, some x between 1 and 2 has f(x) = 0, which guarantees a root and underlies the bisection method for finding solutions.
To make a piecewise function continuous, set the pieces equal at the joining point and solve for the unknown constant. For example, if f(x) = x + 1 for x less than 2 and f(x) = kx − 1 for x at least 2, matching values gives 3 = 2k − 1, so k = 2.
Continuity is the assumption behind many theorems of calculus and it models real quantities like temperature, which cannot jump instantly. Removable discontinuities can be repaired by redefining one value; jumps and asymptotes cannot.
Key ideas
- Limit exists and equals f(a)
- Jumps and holes break continuity
- Polynomials are continuous everywhere
Quick check
Use cases: where Continuity helps
- Homework: review the key idea behind Continuity before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Continuity without notes.
- Visual learning: play with the interactive lab to see how Continuity changes when you adjust its inputs.
- Teaching a friend: explain Continuity in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Continuity?
A function is continuous if you can draw it without lifting the pencil.
What will I learn in Continuity?
Limit exists and equals f(a). Jumps and holes break continuity. Polynomials are continuous everywhere.
Is Continuity part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Continuity lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Continuity?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.