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(vertical asymptote)

Try these different functions so you obtain the idea:

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



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.

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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,


limx→cf(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 cthen 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:

"2" from the left, and"1" from the right

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