Look out for holes, jumps or upright asymptotes (where the duty heads up/down in the direction of infinity).

You are watching: How to tell if a function is continuous without graphing

**Continuous**

Not

Not

**(hole)**

NotContinuous

Not

**(jump)**

NotContinuous

Not

**(vertical asymptote)**

Try these different functions so you obtain the idea:

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

## Domain

Afunctionhasa Domain.

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

So there is a "discontinuity" at x=1

f(x) = 1/(x−1)So f(x) = 1/(x−1) end all genuine Numbers is not continuous

Let"s readjust the domain come **x>1**

**g(x) = 1/(x−1) because that x>1**

So g(x) IS continuous

In other words g(x) go **not** include the worth x=1, so the is **continuous**.

When a role is **continuous in ~ its Domain**, the is a consistent function.

See more: How To Get Hard Nipples All The Time, How Do You Make Nipples Hard All The Time

## More official !

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

A function **f** is continuous when, because that **every** worth **c** in its Domain:

f(c) is defined,

and

*lim***x→c**f(x) = f(c)

"the limit of f(x) together x approaches c equates to f(c)"

The border says:

"as x gets closer and also closer come c**then f(x) it s okay closer and also closer to f(c)"**

**And we have actually to inspect from both directions:**

as x viewpoints c (from left)then f(x) ideologies f(c) | ||

ANDas x viewpoints c (from right)then f(x) approaches f(c) |

**If we get various values from left and also right (a "jump"), climate the border does no exist!**

**And psychic this has to be true for every worth c** in the domain.

## How to Use:

Make certain that, for every **x** values:

**f(x)**is definedand the border at

**x**equates to

**f(x)**

Here room some examples:

### Example: f(x) = (x2−1)/(x−1)forallRealNumbers

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

(x2−1)/(x−1) = (12−1)/(1−1) = **0/0**

So the is **not** a continuous function

Let us readjust the domain:

### Example: g(x) = (x2−1)/(x−1) end the expression xpiecewise function:

It looks choose this:

It is **defined** at x=1, since **h(1)=2** (no "hole")

But at x=1 **you can"t say what the limit is**, due to the fact that there space two completing answers:

so in reality the border does not exist in ~ x=1 (there is a "jump")