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


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Not
Continuous(hole)
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Not
Continuous(jump)
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Not
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.)

Domain

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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

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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

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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,

and

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

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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:

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It looks choose this:

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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")