# Types of discontinuity

A removable discontinuity is a single hole. A jump has different left and right limits. An infinite discontinuity has the function growing without bound.

Maths (university) · Part 02: Limits and continuity · Card 7 of 9 · Last updated 2026-10-19

## Analogy

Three kinds of road damage: a small pothole, a step up, and a cliff edge.

## Example

**Three examples at x = 0 or 1**

(x^2-1)/(x-1) at 1: removable hole.
|x|/x at 0: jump, from -1 to 1.
1/x^2 at 0: infinite, grows without bound.

## In practice

Naming the type tells you whether the function can be repaired by redefining one value.

## Practice

### Flashcards

- **Which discontinuity has different left and right limits?** A jump discontinuity.

### Interview questions

**Describe the three types of discontinuity and give an example of each.**

- Removable: hole, limit exists, value missing
- Jump: left and right limits differ
- Infinite: function grows without bound
- Examples such as (x^2-1)/(x-1), |x|/x, 1/x^2

## Questions people ask

### Can a removable discontinuity be fixed?

Yes. If the limit exists but f(a) is missing or wrong, you define a new function with f(a) set equal to the limit. For (x^2 - 1)/(x - 1), setting the value at 1 to 2 fills the hole and makes the function continuous.

### What is a jump discontinuity?

A jump happens when the left-hand and right-hand limits both exist but are different numbers. The graph suddenly steps from one height to another. The sign function, which gives -1 for negative inputs and 1 for positive inputs, jumps by 2 at zero.

## Related concepts

- [Continuity](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-6)
- [Intermediate value theorem](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-8)
- [Differentiability](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-10)

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