Key takeaways
- f is continuous at a when lim(x→a) f(x) = f(a).
- Three checks: f(a) is defined, the limit exists, and they are equal.
- Breaks come in three kinds: holes, jumps and infinite (vertical asymptote).
- Polynomials are continuous everywhere; 1/x breaks at x = 0.
In this guide
What does continuous mean?
Picture drawing a graph with a pencil. If you never need to lift the pencil near x = a, the function is continuous at a. If there is a hole, a jump or the graph shoots off to infinity, it is discontinuous there.
| Type of break | What it looks like | Example |
|---|---|---|
| Hole | One missing dot in the graph | (x² − 1)/(x − 1) at x = 1 |
| Jump | The graph suddenly leaps up or down | A parking fee that goes from $10 to $15 at the 2-hour mark |
| Infinite | The graph shoots up or down forever | 1/x at x = 0 |
Why does continuity matter?
Many of the most useful results in calculus only work for continuous functions. The intermediate value theorem, which proves an equation has a solution, needs continuity. So do the mean value theorem and the rule that limits can be found by plugging in.
In real life, continuity separates smooth change from sudden change. Temperature over a day changes smoothly. A postage price jumps when a parcel crosses a weight limit.
Continuity is the promise that small changes in x cause only small changes in f(x). No surprises, no teleporting.
The formula: three conditions
| Condition | What to check | Example with x³ at a = 2 |
|---|---|---|
| 1. f(a) is defined | The function has a value at the point (no hole) | f(2) = 8 |
| 2. The limit exists | Left and right limits agree (no jump) | both sides head to 8 |
| 3. They are equal | The limit equals f(a) | 8 = 8 |
This calculator checks f(x) = xⁿ at a point a. For whole-number powers n ≥ 0, xⁿ is a polynomial, which is continuous everywhere, so all three checks pass and the result is f(a) = aⁿ. With a negative power at a = 0 (for example n = −1), it reports a discontinuity: f(0) is undefined and the graph shoots off to infinity. With a fractional power at a negative a (like x^0.5 at −4), it reports that the function isn't defined there. For other functions with holes or jumps, compare the left and right limits by hand.
How to check continuity step by step
- 1Find f(a). Put x = a into the function. If you get a real number, condition 1 holds.
- 2Find the one-sided limits. Work out the limit from the left and from the right. If they match, condition 2 holds. For polynomials, see the limit calculator.
- 3Compare. If the limit equals f(a), the function is continuous at a.
Now test your own power and point below. The calculator runs all three checks and tells you which one fails, if any.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Checking only one side. For piecewise functions, the left and right limits can differ.
- Assuming every formula is continuous. Fractions fail where the bottom is 0, and square roots fail for negative inputs.
- Using negative powers at 0. x⁻¹ = 1/x is not defined at x = 0, so it is not continuous there.
- Confusing 'defined' with 'continuous'. Being defined is only the first of three checks.
Tips and tricks
- Holes can be fixed: if only condition 1 or 3 fails, redefine f(a) as the limit to 'fill the hole'.
- Piecewise functions: check the point where the formula changes, from both sides.
- Next step: continuity is needed before you can take a derivative. See the derivative power rule calculator.
Where you'll use it in real life
- Maths: proving an equation has a solution with the intermediate value theorem.
- Science: modelling smooth quantities like temperature or distance travelled.
- Economics: spotting jumps in tax brackets or price tiers.
- Engineering: making sure control signals don't suddenly jump and damage equipment.
Quick summary
A function is continuous at a point when it is defined there, its limit exists there, and the two are equal. Check all three, and watch for holes, jumps and infinite breaks. Use the calculator above to test xⁿ at any point with every condition shown.
Worked example
Your teacher asks you to show that f(x) = x³ is continuous at x = 2 by checking all three conditions. Here n = 3 and a = 2.
- 01f(x) = x^3, checking continuity at x = 2
- 02
- 03Condition 1: f(2) is defined
- 04 f(2) = 2^3 = 8 ✓
- 05
- 06Condition 2: lim[x→2] f(x) exists
- 07 x^3 is a power function, continuous on its whole domain, so lim[x→2] x^3 = 8 ✓
- 08
- 09Condition 3: lim[x→2] f(x) = f(2)
- 10 8 = 8 ✓
- 11
- 12Conclusion: f(x) = x^3 is CONTINUOUS at x = 2
These numbers are pre-filled in the calculator above.
Frequently asked questions
What are the three conditions for continuity?
f(a) must be defined, the limit of f(x) as x → a must exist, and the limit must equal f(a). If any one fails, the function is discontinuous at a.
Are all polynomials continuous?
Yes. Every polynomial, like x³ − 2x + 1, is continuous at every real number, so you can find limits by plugging in.
Is 1/x continuous?
It is continuous at every x except 0. At x = 0 it is not defined, and the graph shoots to +∞ on the right and −∞ on the left.
What is a removable discontinuity?
It is a hole: the limit exists, but f(a) is missing or has the wrong value. You can remove it by defining f(a) to equal the limit.
