Continuity

As an informal but intuitive way of describing continuity, we can think of a continuous function as a function which can be drawn without lifting the pen.

This does not paint the whole picture, and we also have to remember that all figures we can draw without lifting the pen are not functions. Still, this criterion can help us determine which points make a function discontinuous.

However, to put the concept of continuity in the rigorous framework of calculus, we must state a more formal definition:

For the second criterion to hold, we have that must approach the same value when approaches from both above and below.

Furthermore, to fulfill the other two, this number must be the function's value at , which then has to be defined.

To see what continuity at points look like in practice, let's view a couple of examples:

Look at the function:

at the point .

Since division is undefined for 0 as denominators, is not defined. Also, considering that:

while:

No limit exist of as goes to .

Consequently, none of the three criteria are fulfilled, and we conclude that is discontinuous at . As a side note, the function is continuous for all other values of .

Now let's look at a different function, at a different point :

Clearly, is defined at and so the first criterion is satisfied. However, also here the limits of the function are different as approaches from above and from below.

Therefore,

does not exist, which breaks the second of the requirements and makes discontinuous at .

It is about time that we now look at a function that is continuous at all points in :

We will consider it at the point , where continuity might not be obvious.

Firstly, , so the function is defined for this point.

Secondly:

gives us existence of the limit at the point.

Lastly, the limit not only exists at the point, but also happen to be the same as the function's value:

With all three criteria satisfied, we conclude that is continuous at .

A function is continuous in a point if the limit as approaches coincides with the function value, that is:

Assume that the function is continuous in . Ok, so what?

Most theorems about continuous functions require that the function be continuous on an interval.

The fact that is continuous in isn't that useful per se. However, it's part of the definition of continuity on an interval.

The function is continuous, as a whole, if it's continuous on all points for which it's defined.

Now we turn to the question of which functions are continuous.

The basic functions are all continuous. They include:

As a rule of thumb, whatever you do with the functions above will give a continuous function.

However much you try - summing, dividing, bending and twisting - you won't be able to put together a discontinuous function.

Note that there is one exception to the rule above, and that is whenever we have a function that is zero somewhere on the axis then that function will cause a discontinuity at that .

It turns out that:

are all continuous.

By using these rules, you can tell that a monstrous function like

is continuous.

Continuous functions are practical. Many theorems which are helpful in handling functions require that the function is continuous, else it may give some rather weird results.

Therefore, if a function is undefined at a point, we sometimes wish to modify it so that it becomes continuous. The modified function is not the same, but for practical purposes it may not matter.

As an example, let's revisit a friend:

This function is continuous everywhere, except at .

We can extend the function definition to be:

Then, is continuous, and we are free to use for example the intermediate value theorem and the max-min theorem when dealing with the function!

Check out this function graph:

Let's say that the function is only defined on the closed interval .

By looking at the graph, we conclude that the function has a maximum value of and a minimum value of .

Put differently, there exists a maximum value and a minimum value.

The prerequisites are important. They're not just some technical details some mathematician came up with to irk undergraduates.

First, the interval must be closed. Take the function , defined on the open interval .

There isn't any maximum, since we can always move one tiny step to the right to increase the function value. Similarly, there isn't any minimum.

Second, the function must be continuous. Say hello to our old friend

Even if we consider a closed interval, like , there isn't any maximum, nor any minimum.

The function graph goes a bit crazy around , since

In this course, we mostly deal with functions which are continuous on almost all of their domain (i.e. their allowed input values).

Today, we are going to look at a very neat theorem which can only be used for continuous functions. It's called the intermediate value theorem, and informally put, it says:

If is less than , and is greater than , then somewhere in the interval .

Trying to with one pen stroke get from to in the graph below might convince you of this fact.

Say we have a function like this one:

Where can we find its roots? Or, put differently, where does its graph cut the -axis?

By the help of the intermediate value theorem, we can answer this question. Just use .

Maybe you have heard that there is no efficient analytical method for solving polynomials of higher degree. As soon as there appears an or something even worse in a polynomial, then we need to use our special toolbox of tricks. These tricks only work in some cases.

However, we can always solve a polynomial equation numerically. A numerical solution is an approximation, which is often found by taking gradual steps towards a good enough answer.

In this case, I happen to know that at , and that at , .

As the function is continuous, we then know that the graph must make its way from to in the interval .

So, there must be at least one root in this interval!

Finding a root can be done by cutting the interval in half and, each time we cut it in half, check which interval has one -value below zero and which one has above zero. Rather practical, right?

Finally, we are ready for the theorem:

Note that it's even stronger than what we have said so far: the theorem promises that all values between and will be reached.