A function is continuous at a point if it is defined there, the limit exists, and the two agree.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
All three, not any one
The hole example had a limit but no value. That is enough to break continuity.
Types of break
A removable hole, a jump where the one-sided limits differ, and an infinite break at an asymptote.
The Intermediate Value Theorem
A continuous function that goes from below a value to above it must pass through it.
Three conditions, all required
f(a) must exist, the limit as x approaches a must exist, and the two must be equal. Failing any one breaks continuity, and each failure produces a visibly different kind of break.
The informal picture
A continuous function can be drawn without lifting the pencil. That intuition is reliable for the functions met in a first course, and the three-condition definition is what makes it precise.
What is continuous
Polynomials are continuous everywhere. Rational functions are continuous except where the denominator vanishes. Roots, exponentials, logs and trigonometric functions are continuous throughout their domains.
Why calculus cares
The major theorems — intermediate value, extreme value, mean value, and the fundamental theorem — all require continuity. It is the hypothesis that makes those results true, not a technicality attached to them.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Priya Apply the IVT
f(x) = x³ − x − 1 is continuous. Priya shows it has a root between 1 and 2.
- Step 1
She computes f(1) = 1 − 1 − 1 = −1, which is below zero.
Step 4: Your Turn
Practice makes it stick.
The Value
Problem 1 of 2
f(x) = x³ − x − 1. What is f(2)?
The Conditions
Problem 2 of 2
How many conditions must hold for continuity at a point?
Breaks and Bridges
1 of 4
f(x) = x² − 3. What is f(2)?
2 of 4
f(x) = x² − 3. What is f(1)?
3 of 4
y = 1/(x − 4). At what x is there an infinite discontinuity?
4 of 4
Can a function have a limit at a point where it is undefined?
Step 5: Quick Check
Show what you know.
Question 1 of 2
f(x) = x³ − 4x. What is f(3)?
Question 2 of 2
What does the Intermediate Value Theorem guarantee?
What You Learned
- Continuity needs a value, a limit, and their agreement.
- Breaks come as holes, jumps, or asymptotes.
- A continuous function cannot skip a value.