Function is continuous at , if it is interpretable at , and it has a limit that is finite at , and most importantly:

Let's see an example.

Is the following function continuous at 3?

Well, those who have the ability to see things, know it right away that it isn't,

because this function jumps at 3, and jumping is very bad for a function's continuity.

Let’s see how we could get this without drawing.

We compute the limit,

and then the value of the function,

and if they are equal, then the function is continuous, if they are not, then it is not continuous.

They are not equal, so the function is not continuous at 3.

Not continuous at 3, but it could be made continuous.

All we have to do is modify this.

And now it is continuous.

It is not that simple at 4.

It would be rather difficult to make this function continuous.

In fact, impossible.

Function can be made continuous at if it has a value at

and it has a finite limit at .

Here is another function. The task is to find the values of parameters and such that the function is continuous at 2 and at 3.

The drawing is only black magic, again.

Let's see the limits. Let's start with 2.

Then let's see what it is at 3.

We have successfully factorized this before.

And we have even simplified it.

cannot be defined such that the function is continuous at 3.

So, this function can be made continuous at 2 if A=4, but it cannot be made continuous at 3.

Let's see a third function, too.

Find out whether it could be made continuous at x=1 and at x=3.

If we take a look at the figure, we can see that at 1, the limit is finite, but at 3, it is infinite.

Consequently, the function can be made continuous at 1, but not at 3.

Let’s see how we can get this without drawing.

The limit is finite, so the function can be made continuous.

Namely, by doing this:

There is no limit, so sadly, it cannot be made continuous at 3.