If you're a cellist, you can slide your finger along the string to increase the pitch. As long as you don't lift the bow, it'll sound as if the notes hang together.
Unless you're some kind of Mozart, you can't discern the individual notes.
In contrast, a piano player can only play notes corresponding to the white and black keys. The notes will flow together in the same way and you won't get the same feeling of continuity.
There are some exotic functions whose graphs make leaps or have holes. For example, have a look at the function graph of :
As approaches , the becomes either super big or super tiny, depending on whether approaches from the right or the left. This function is discontinuous.
A function is continuous if its graph can be drawn without lifting the pen
Other functions, such as , and , behave more regularly - they have graphs which can be drawn without lifting the pen. These are called continuous.
Imagine holding two pens, one in each hand. Let the pens approach some point on the graph. For example, consider this piecewise defined function:
At the point where , the function graph suddenly jumps from to . If the function were continuous at , the pen nibs would meet each other.
To qualify as a continuous function, we require that:
Continuity at a point
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.
A function is discontinuous at points where we are forced to lift the pen when drawing its graph
However, to put the concept of continuity in the rigorous framework of calculus, we must state a more formal definition:
Let be a real-valued function, and let .
Then is said to be continuous at if and only if:
Unless these three hold, we say that the function is discontinuous at .
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:
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.
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.
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 .
Continuity on an interval
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.
A function is continuous on an interval if it's continuous in every point on that 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:
Rational functions (one polynomial divided by another polynomial)
Exponentials and logarithms
Trigonometric functions and inverse trigonometric functions
Determining whether afunction is continuous
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:
Sums of continuous functions
Products of continuous functions
Quotas of continuous functions, except for when the denominator is zero on the axis
Compositions (like of continuous functions
are all continuous.
By using these rules, you can tell that a monstrous function like
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!
The max-min theorem
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.
If a function is continuous on a closed interval , then assumes a maximum and a minimum value on the closed interval .
The prerequisites are important. They're not just some technical details some mathematician came up with to irk undergraduates.
The interval must be closed and the function must be continuous
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
The intermediate value theorem
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.
An illustrative example
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!
If takes both positive and negative values on an interval, then there must be a root within that 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?
The intermediate value theorem
Finally, we are ready for the theorem:
The intermediate value theorem
Let be continuous on . Then, f will take on all values between and .
Note that it's even stronger than what we have said so far: the theorem promises that all values between and will be reached.