Motivation and derivation[Bearbeiten]
Consider the limit . In school, this limit would be calculated as follows:
The sign function
Intuitively, this calculation makes sense: if , then should hold. But can we really argument like that? Why should we be allowed to "pull" the limit inside the function brackets? Let us consider another example: the sign function , which is returning the sign of :
Since there is:
So . This simple example shows that the limit may not simply be pulled into the function brackets. The function plot shows, why this worked with , but not with . For , the sequence converges to , when taking the limit :
The sign function has a "jump" in the graph at and hence, the sequence does not converge to :
We note: There are certain functions, where the limit may be pulled into the brackets, wile for other functions this may not always work.
Jumps and continuity[Bearbeiten]
The reason why we cannot pull the limit into the sign function , is because its graph has a "jump" at . Let us now try to figure out, why pulling the limit into the brackets does not work, if the graph has a jump at the limit of the argument sequence. We assume to have the following graph:
When approaching from the left, the function values will also approach . So if the sequence of arguments consists only or almost only (meaning: with finitely many exceptions) of real numbers smaller or equal , then we may pull the limit inside the brackets. However, this fails if infinitely many elements bigger than appear in the sequence of arguments. Their function values will not approach , as the graph is having a jump at , when looking to the right. This jump causes a minimal distance which function values in the proximity and right of cannot fall below. The jump at prevents us from being able to pull the limit into the brackets for some sequences of arguments.
The same happens if the graph has a jump at in the left-handed direction:
Here, pulling the limit in fails whenever the sequence of arguments contains infinitely many elements smaller than . Function values left of will not approach due to the jump.
Mind that the situation may be different if is not defined at the jump:
Here, the expression does not make sense, since is not defined at . So we do not need to consider, whether pulling in the limes is allowed there. For all other real numbers, the graph of is continuous. We observe: A jump does only cause discontinuity of a function , if is even defined at the position of the jump.
Transition to a formal definition[Bearbeiten]
Let us take again a function with a jump at . When approaching the argument from one side, a certain minimal distance between and will always be kept. This minimal distance between and is caused by the jump at . When approaching from the other side, the -values will come arbitrarily close to (provided there is no second jump).
For -values that should approach arbitrarily (=„infinitely“) close to , we can use the notion of a sequence. To do so, we treat the -values as a sequence , converging to . The notion of a sequence is useful, as we need usually need infinitely many -values for the approach and sequences are containing infinitely many elements as well.
Now, let us assume that we approach from the side, where our obey a minimal distance to for in the vicinity of . This minimal distance will be sustained in the limit process . In case that exists, we can therefore be sure that .
However, in our example we may also choose the sequence such that . This will, for instance, be the case when is approached by our from the other side. So in order to have , we cannot make an arbitrary choice of our sequence converging to . But we certainly know that there are sequences , for which does not tend towards .
Derivation of the sequence criterion[Bearbeiten]
Let us recap what we found so far:
Whenever a function has a jump at , then we may find at least one sequence of arguments with but .
So if there is a jump at then we may not pull the limit inside the brackets (i.e. ) for at least one sequence of arguments converging to . Now, our intuition tells us that a function is discontinuous at some argument , whenever its graph has a jump there. Therefore, we may define:
The graph of a function is discontinuous at some argument , if there is at least one sequence of arguments with but .
In order to find the corresponding definition for continuity at , we just need to take the negation of the above definition for discontinuity. That means, we may understand continuity as absence of a jump, where the exact definition reads:
A function is continuous at , if for all sequences of arguments in with we may pull the limit inside the brackets:
So we memorize: in case a function is continuous at some argument , we may pull the limit inside the brackets for any sequence of arguments convrging to . This is the sequence criterion of continuity. The entire function is considered to be continuous, if it is continuous at each argument in its domain of definition. The definition of continuity for a function therefore reads:
A function is continuous, if for all convergent sequences , there is . Here, all elements of the sequence and the limit must be inside the domain of , since otherwise the equation would make no sense.
So let us recap: For continuous functions , we may always pull the limit inside the brackets – regardless of the limit of the sequence of arguments. For instance, the exponential function is continuous, so we may always pull the limit inside the brackets of this function. The sign function is discontinuous at , so we may not pull the limit inside the brackets, if the sequence of argument converges to zero.
In the above section, we already learned the definition of continuity. Let us formalize this:
Definition (Sequence criterion of continuity for a single argument)
A function with is continuous at an argument , if for all squences with and there is:
A function is said to be continuous if it is continuous at each argument:
Definition (Sequence criterion of continuity)
A function with is continuous, if for all and all sequences with and there is:
So continuity guarantees us that we are allowed to pull the limit inside the brackets. This may simplify calculations of limits tremendously - and indeed, continuity is a concept one may encounter frequently when calculating limits.
Consequences of this definition[Bearbeiten]
We just found a formal definition of continuity by using our intuition of jumps. This definition turned out to be useful over the years and is therefore accepted in the mathematical community as the definition of continuity. And we will use it for the following considerations as well. So if we want to decide whether a function is continuous, we will replace our first intuition by the sequence criterion which we just found. This will also have some side effects. We consider the topological sine function:
Its graph is given by:
Our first intuition is not suitable for deciding, whether this function is continuous at , since it is oscillating here infinitely many times with the period of oscillation tending to zero. The graph itself does not look as if there would be a jump. However, using the sequence criterion of continuity, we may show that the topological sine function is discontinuous at , which corresponds to the infinitely fast oscillations (Exercise).
As a byproduct of our definition, we found a type of discontinuity differing from a jump. It is also called essential discontinuity . And we need to accept these kind of discontinuities, if we want to use the sequence criterion as a definition for discontinuities.
Please note that this is one of the standard definitions in mathematics. So denying it or looking for an alternative would certainly make us some lonely outsiders in the world of mathematicians!
Examples for the sequence criterion[Bearbeiten]
Quadratic functions are continuous[Bearbeiten]
Exercise (Continuity of quadratic functions)
Show that the quadratic function is continuous.
Proof (Continuity of quadratic functions)
Let . Consider any sequence , converging to . There is
So we may pull the limit inside the brackets for the quadratic function and hence, it is continuous.
Application of the sequence criterion[Bearbeiten]
We have just proven that the quadratic funtion is continuous. So we may pull the limit inside the brackets without further thinking about it. This is a nice thing about continuity, as the following example underlines:
Exercise (Application of the sequence criterion)
Solution (Application of the sequence criterion)
We may use continuity of the quadratic function by pulling the limit inside the brackets. This is allowed by the sequence criterion. And it yields us:
Common sketches of proofs[Bearbeiten]
Proofs of continuity using the sequence criterion[Bearbeiten]
In order to prove continuity of a function at some , we need to show that for each sequence of arguments converging to , there is . A proof for this could schematically look as follows:
Let be a function defined by and . In addition, let be any sequence of arguments satisfying . Then, there is:
In order to prove continuity for the function (for all arguments in its domain of definition), we need to slightly adjust that scheme:
Let be a function defined by and let be any element of the domain of definition for . In addition, let be any sequence of arguments satisfying . Then, there is:
Proving discontinuity using the sequence criterion[Bearbeiten]
In order to show that a function is discontinuous at using the sequence criterion, we need to find one specific sequence of arguments with for all which is converging to , such that the sequence of function values does not converge to . So there shall be but . In order for to hold, there are two cases to be distinguished:
- The sequence of function values diverges.
- The sequence of function values converges, but its limit is not .
Therefore, a proof of discontinuity using the sequence criterion could take the following form:
Let be a function defined by . This function is discontinuous at for the following reason: We take the sequence with and all these elements are inside the domain of definition for . This sequence converges to
However, there is . And instead, there is ...Proof that diverges or that the limit of exists and is different from ...
Equivalence to the epsilon-delta criterion[Bearbeiten]
Now, we have two definitions of continuity: the epsilon-delta and the sequence criterion. In order to show that both definitions describe the same concept, we have to prove their equivalence. If the sequence criterion is fulfilled, it must imply that the epsilon-delta criterion holds and vice versa.
epsilon-delta criterion implies sequence criterion[Bearbeiten]
Theorem (The epsilon-delta criterion implies the sequence criterion)
Let with be any function. If this function satisfies the epsilon-dela criterion at , then the sequence criterion is fulfilled at , as well.
How to get to the proof? (The epsilon-delta criterion implies the sequence criterion)
Let us assume that the function satisfies the epsilon-delta criterion at . That means:
For every , there is a such that for all with .
We now want to prove that the sequence criterion is satisfied, as well. So we have to show that for any sequence of arguments converging to , there also has to be . We therefor consider an arbitrary sequence of arguments in the domain with . Our job is to show that the sequence of function values converges to . So by the definition of convergence:
For any there has to be an such that for all .
Let be arbitrary. We have to find a suitable with for all sequence elements beyond that , i.e. . The inequality seems familiar, recalling the epsilon-delta criterion. The only difference is that the argument is replaced by a sequence element - so we consider a special case for . Let us apply the epsilon-delta criterion to that special case, with our arbitrarily chosen being given:
There is a , such that for all sequence elements fulfilling .
Our goal is coming closer. Whenever a sequence element is close to with , it will satisfy the inequality which we want to show, namely . It remains to choose an , where this is the case for all sequence elements beyond . The convergence implies that gets arbitrarily small. So by the definition of continuity, we may find an , with for all . This now plays the role of our . If there is , it follows that and hence by the epsilon-delta criterion. In fact, any will do the job. We now conclude our considerations and write down the proof:
Proof (The epsilon-delta criterion implies the sequence criterion)
Let e a function satisfying the epsilon-delta criterion at . Let be a sequence inside the domain of definition, i.e. for all coverging as . We would like to show that for any given there exists an , such that holds for all .
So let be given. Following the epsilon-delta criterion, there is a , with for all close to , i.e. . As converges to , we may find an with for all .
Now, let be arbitrary. Hence, . The epsilon-delta criterion now implies . This proves and therefore establishes the epsilon-delta criterion.
Sequence criterion implies epsilon-delta criterion[Bearbeiten]
Theorem (The sequence criterion implies the epsilon-delta criterion)
Let with be a function. If satisfies the sequence criterion at , then the epsilon-delta criterion is fulfilled there, as well.
How to get to the proof? (The sequence criterion implies the epsilon-delta criterion)
We need to show that the following implication holds:
This time, we do not show the implication directly, but using a contraposition. So we will prove the following implication (which is equivalent to the first one):
Or in other words:
So let be a function that violates the epsilon-delta criterion at . Hence, fulfills the discontinuity version of the epsilon-delta criterion at . We can find an ,such that for any there is a with but . It is our job now to prove, that the sequence criterion is violated, as well. This requires choosing a sequence of aguments , converging as but .
This choice will be done exploiting the discontinuity version of the epsilon-delta criterion. That version provides us with an , where holds (so continuity is violated) for certain arguments . We will now construct our sequence exclusively out of those certain . This will automatically get us .
So how to find a suitable sequence of arguments , converging to ? The answer is: by choosing a null sequence . Practically, this is done as follows: we set . For any , we take one of the certain for as our argument . Then, but also . These make up the desired sequence . On one hand, there is and as , the convergence holds. But on the other hand , so the sequence of function values does not converge to . Let us put these thoughts together in a single proof:
Proof (The sequence criterion implies the epsilon-delta criterion)
We establish the theorem by contraposition. It needs to be shown that a function violating the epsilon-delta criterion at also violates the sequence criterion at . So let with be a function violating the epsilon-delta criterion at . Hence, there is an , such that for all an exists with but .
So for any , there is an with but . The inequality can also be written . As , there is both and . Thus, by the sandwich theorem, the sequence converges to .
But since for all , the sequence can not converge to . Therefore, the sequence criterion is violated at for the function : We have found a sequence of arguments with but .
Continuity of the absolute function[Bearbeiten]
Exercise (Continuity of the absolute function)
Prove continuity for the absolute function.
Proof (Continuity of the absolute function)
Let with be the absolute function. Let be a real number and a sequence converging to it . In chapter „Grenzwertsätze: Grenzwert von Folgen berechnen“ we have proven the absolute rule, stating that , whenever there is . Hence:
This proves continuity of the absolute function by the sequence criterion.
Discontinuity of the topological sine function[Bearbeiten]
Exercise (Discontinuity of the topological sine function)
Prove discontinuity of the following function:
How to get to the proof? (Discontinuity of the topological sine function)
Discontinuity of means that this function has at least one argument where is discontinuous. For each , is equal to the functionn in a sufficiently small neighbourhood of . Since is just a composition of continuous functions, it is continuous itself and therefore, must be continuous for all , as well. So we know that the discontinuity may only be situated at .
In order to prove that is discontinuous at , we need to fincd a sequence of arguments converging to but with . To find such a function, let us take a look at the graph of the function :
In this figure, we recognize that takes any value between and infinitely often in the vicinity of . So, for instance, we may just choose such that is always . This guarantees that - and actually any other real number between and in place of would do the job. But we need to make a specific choice for , and is a very simple one. In addition, we will choose to converges to zero from above.
The following figure also contains the sequence elements beside our function . We may clearly see that for the sequence of function values converges to , which is different from the function value :
But what are the exact values of these for which we would like to have To answer this question, let us resolve the equation for :
So for each with , we have . In order to get positive converging to zero from above, we may for instance choose . In that case:
And we have seen that . So we found just a sequence of arguments , which proves discontinuity of at .
Proof (Discontinuity of the topological sine function)
Let with for and . We consider the sequence defined by . For this sequence:
And there is:
Hence, although . This proves that is discontinuous at and therefore it is a discontinuous function.