More About Continuity

Calculus
Section 2.5

Define the following in your own words and give an example for each.

1. Removable discontinuity

2. Infinite discontinuity

3. Jump discontinuity

 

A function is continuous when its graph is a single unbroken curve that you could draw without lifting your pen from the paper.

Here is a continuous function:

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


So what is not continuous (also called discontinuous) ?



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

Example: 1/(x−1)
At x=1 we have:

1/(x−1) = 1/0 = undefined

So there is a "discontinuity" at x=1


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

Let's change the domain to x>1



So g(x) IS continuous

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

We can define continuous using Limits:
A function f is continuous when, for every value c in its Domain:

f(c) is defined, and lim f(x) = f(c) as x→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.



1. Removable discontinuity

What Are Holes?
Another way we can get a removable discontinuity is when the function has a hole. A hole is created when the function has the same factor in both the numerator and denominator. This factor can be canceled out but needs to still be considered when evaluating the function, such as when graphing or finding the range. When dealing with a function like this, there will be some point where the function is undefined. Look at this function, for example.

f(x)=(x^2-4x)/(x-4)........factor numerator
f(x)=(x(x-4))/(x-4)
This function has the factor x - 4 in both the numerator and denominator. What happens at the point x = 4? Let's see.

f(x)=(4(4-4))/(4-4)=(4*0)/0 =0/0 =>the function is undefined

In the graphs below, there is a hole in the function at x=a. These holes are called removable discontinuities.


2. Infinite discontinuity

If the function doesn't approach a particular finite value, the limit does not exist. This is an infinite discontinuity.

The following two graphs are also examples of infinite discontinuities at x=a. Notice that in all three cases, both of the one-sided limits are infinite.


3. Jump discontinuity
The function is approaching different values depending on the direction x is coming from. When this happens, we say the function has a jump discontinuity at x=a.

The graph of f(x) below shows a function that is discontinuous at x=a.

In this graph, you can easily see that lim(−f(x))=L as x→a and lim(+f(x))=M as x→a.

 

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