Contents of this Calculus 1 tutorial:

Discontinuities, Continuous, right side limit, left side limit, Removable discontinuity, Jump discontinuity, Infinite discontinuity, Oscillating discontinuity.

In the previous slides we discussed when a function is continuous. We also discussed that if it is not continuous, then it can have various types of discontinuities.

Now we will try to make a list of the possible types of discontinuities.

TROUBLES IN THE FUNCTION’S LIFE

If there is a finite limit at a, and it is the same as the value of the function, then the function is continuous.

If there is a finite limit at a, but it is not the same as the value of the function, then the function has a removable discontinuity.

If the left and right-side limits at a are two different numbers, then the discontinuity is not removable. This is called a jump discontinuity.

If the left and right-side limits are not even finite at a, then the function has an infinite discontinuity.

There are some rather pathological cases, where the right or left-side limits don’t even exist.

oscillating discontinuity

removable

jump discontinuity

Here comes a function:

The question is whether this function is continuous at x=2.

Well, based on the figure, it is not.

Let’s see how we can decide without the graph.

The value of the function:

We substitute 2.

Here, we mustn’t. Here we can.

Now, let's see the limit.

The function has two parts, and the two parts are joined right at 2, therefore:

At the left-side limit we have the left-side function,

and at the right-side limit, the right-side function.

So the limit is quite schizophrenic: from one side it is 11, but from the other side it is 7.

Well, this is a bad omen for the continuity of the function.

Since

therefore there is no limit, so the function is not continuous at 2.

The type of the discontinuity, based on our list, is a jump.

But, the left-side limit and the value of the function are the same,

so the function is left-continuous.

If we make a minor modification, and place the equality here,

then the right-side limit will be equal to the value of the function.

In this case, the function is right-continuous.

Let’s see another function.

Could the value of A given such that the following function is continuous

at x=1?

Let's see the graph.

Well, we have no idea what these functions look like,

but it actually doesn’t matter.

All we need to know, which is to the left of 1,

and which is to the right of 1.

Perfect.

Now let's get down to business.

The value of the function:

The limit is a bit schizophrenic, again.

From the left, we look at the limit of the left-side function

The right-side limit and the value of the function are equal.

So the function is definitely right-continuous.

And if the two limits are the same,

then the function is continuous at 1.

Finally here comes one more.

We don’t care much about the looks of this function either.

All we need to know, which is to the left of 3,

which is to the right of 3,

and what happens if x=3.

The value of the function:

The limit:

The right-side limit is sadly, infinite, so the function cannot be right-continuous.

Discontinuities, Continuous, right side limit, left side limit, Removable discontinuity, Jump discontinuity, Infinite discontinuity, Oscillating discontinuity.

Calculus 1 tutorial

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