# Continuity

A function is continuous at a if f(a) exists, the limit at a exists, and the two are equal. You can draw its graph without lifting your pen.

Maths (university) · Part 02: Limits and continuity · Card 6 of 9 · Last updated 2026-10-19

## Analogy

A road with no potholes or gaps: you can drive across without stopping.

## Example

**f(x) = (x^2 - 1)/(x - 1)**

At x = 1 it is undefined (0/0).
For x not equal 1: f(x) = x + 1.
Limit at 1 is 2.
Define f(1) = 2 and it is continuous.

## In practice

Continuous functions behave predictably, and many big theorems need continuity to work.

## Practice

### Flashcards

- **What are the three conditions for continuity at a?** f(a) is defined, the limit exists, and the limit equals f(a).

### 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

### What are the three conditions for continuity at a point?

First, f(a) must be defined. Second, the limit of f(x) as x approaches a must exist. Third, the limit must equal f(a). If any one fails, the function is discontinuous at that point and the graph has a hole, jump or break.

### Are polynomials continuous everywhere?

Yes. Polynomials, sine, cosine and exponential functions are continuous at every real number. Rational functions such as 1/x are continuous wherever they are defined, so they break only where the denominator is zero. Sums and products of continuous functions are also continuous.

## Related concepts

- [Types of discontinuity](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-7)
- [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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