# Continuity

A function is continuous when its graph is a single unbroken curve ...

... that you could draw without lifting your pen from the paper.

- 10 synonyms of continuity from the Merriam-Webster Thesaurus, plus 10 related words, definitions, and antonyms. Find another word for continuity. Continuity: uninterrupted or lasting existence.
- Continuity, in mathematics, rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable—say x —is associated with a value of a dependent variable—say y.
- Offering solutions to complex problems worldwide, Continuity Global Solutions leverages extensive experience and exceptional client service to support projects in austere and remote environments. Continuity professionals apply Risk Management and Project Management strategies to every project and mission, ensuring our clients can focus on what.

Continuity definition: Continuity is the fact that something continues to happen or exist, with no great. Meaning, pronunciation, translations and examples.

That is not a formal definition, but it helps you understand the idea.

Here is a continuous function:

## Examples

So what is **not continuous** (also called **discontinuous**) ?

Look out for holes, jumps or vertical asymptotes (where the function heads up/down towards infinity).

Not Continuous | Not Continuous | Not Continuous |

(hole) | (jump) | (vertical asymptote) |

Try these different functions so you get the idea:

(Use slider to zoom, drag graph to reposition, click graph to re-center.)

## Domain

A function has a Domain.

In its simplest form the domain is all the values that **go into** a function.

We may be able to choose a domain that makes the function continuous

### Example: 1/(x-1)

At x=1 we have:

So there is a 'discontinuity' at x=1

f(x) = 1/(x-1)

So f(x) = 1/(x-1) over **all Real Numbers** is NOT continuous

Let's change the domain to **x>1**

g(x) = 1/(x-1) for **x>1**

So g(x) IS continuous

In other words g(x) does **not** include the value x=1, so it is **continuous**.

When a function is **continuous within its Domain**, it is a continuous function.

## More Formally !

We can define **continuous** using Limits (it helps to read that page first):

A function **f** is continuous when, for **every** value **c** in its Domain:

f(c) is defined,

and

*lim***x→c***f(x) = f(c)*

*'the limit of f(x) as x approaches c equals f(c)*'

The limit says:

'as x gets closer and closer to c

then f(x) gets closer and closer to f(c)'

And we have to check from both directions:

as x approaches c (from left) then f(x) approaches f(c) |

AND as x approaches c (from right) then f(x) approaches f(c) |

If we get different values from left and right (a 'jump'), then the limit does not exist!

And remember this has to be true for every value **c** in the domain.

## How to Use:

Make sure that, for all **x** values:

**f(x)**is defined- and the limit at
**x**equals**f(x)**

Here are some examples:

### Example: f(x) = (x^{2}-1)/(x-1) for all Real Numbers

### Continuity Of Care

The function is **undefined** when x=1:

(x^{2}-1)/(x-1) = (1^{2}-1)/(1-1) = **0/0**

So it is **not** a continuous function

Let us change the domain:

### Example: g(x) = (x^{2}-1)/(x-1) over the interval x<1

**Almost** the same function, but now it is over an interval that does **not** include x=1.

So now it **is** a continuous function (does not include the 'hole')

### Example: How about this piecewise function:

that looks like this:

It is **defined** at x=1, because **h(1)=2** (no 'hole')

But at x=1 **you can't say what the limit is**, because there are two competing answers:

- '2' from the left, and
- '1' from the right

so in fact the limit does not exist at x=1 (there is a 'jump')

And so the function is **not continuous**.

But:

### Example: How about the piecewise function absolute value:

At x=0 it has a very pointy change!

But it is still **defined** at x=0, because **f(0)=0** (so no 'hole'),

And the limit as you approach x=0 (from either side) is also **0** (so no 'jump'),

So it is in fact ** continuous**.

### Continuity Definition

(But it is not differentiable.)