Definition of the dorsogluteal point if

Consequence criterion of continuity: Consequence continuity - Serlo "Math for non-freaks"

Motivation and derivation

First examples [edit]

Look at the limit . At school, this limit would be calculated as follows:

The Signum function

This calculation makes intuitive sense: If , then should be. But can we argue that way? Is it allowed to pull the Limes into the function? For this, consider the sign function , which is the sign of returns:

Because of applies:

So is . This shows that the Limes cannot easily be drawn into a function. In the function plot you can see why this is at , but not at is possible. At namely, the sequence converges against , if goes:

With the sign function there is a jump in the graph at , and therefore the sequence converges not against :

We note: There are functions in which the Limes can be drawn in, and functions in which it does not (always) work.

Jump points and continuity [edit]

We realize that this is why we are not in the Limes can drag in because the graph has the sign function at the point "Takes a leap". We now consider why it is not possible to draw in the limit if the graph makes a jump at the limit of the argument sequence. Let's assume that has the following graph:

If we look at each other from the left approach, then the function values ​​also approach at. So if the argument sequence is almost exclusively (read: with a finite number of exceptions) only real numbers less than or equal exists, we can pull the Limes into it. However, this fails if there are infinitely many numbers greater than than in the argument sequence occur. Namely, their function values ​​do not approach each other because the graph is at makes a jump in the right direction. Because of the jump, there is a minimum distance that the function values ​​are near and to the right of not fall below. The leap of the function at the point prevents the Limes from being drawn into every sequence of arguments.

Something similar happens when the graph is at has a jump in the left direction:

Here the inclusion of the Limes fails if the argument sequence is infinitely many numbers smaller than owns. The function values ​​to the left of do not approach each other because of the jump at.

Otherwise the situation is different if is not defined at the jump point:

Here is the expression no sense because the function at the point is not defined. Therefore we do not have to consider whether the Limes can be drawn into there. In all other places is the graph of continuously and therefore steadily. We see: A jump only makes a function discontinuous if the function is defined at the jump point.

Transition to the formal definition [edit]

Let's take a function with jump point in point . If you look at the argument Approaching from one side, there will be a certain distance between and never fallen below. This minimum distance between and was through the jump at the point caused. If you look at the other side approaching, the go Values ​​as close to approach (provided that there is no second jump point here).

For Values ​​that any (= "Infinite") Near at we can use the term sequence. For this we describe the Values ​​as a result , against converges. The use of the term sequence also makes sense because we often have an infinite number for the approximation Need values ​​and sequences also have an infinite number of terms.

Let us now assume that we are from the side approaching ours across from some minimum distance near not fall below. This remains in the Limes receive. If exists, so we know for sure that is.

Now we have only looked at the approach from the side where