Every function has a rule

Features of functions¶

Functions can be divided based on various properties. Important properties that are of importance are briefly summarized in the following section. [1]

Definition and value set¶

The amount of possible values ​​that the output variable ("variable") can assume is called a definition set . The amount of values ​​that make up the function is called accordingly as results, as a set of values .

Sometimes individual values ​​or intervals have to be excluded from the definition set in order to ensure that the function always behaves clearly.


Individual values ​​excluded from the definition set are called definition gaps. If, on the other hand, intervals have to be excluded from the definition set, the remaining definition set is often referred to as the domain and it is also given as the union of intervals.

In the following, only “real-valued” functions are examined, that is, rules that define the real values ​​of an (independent) variable also real values ​​of the (from dependent) variables to assign. Unless there are further restrictions, this applies here .[2]

Representations of functions¶

In general, functions can be represented in three different ways:

  • as a table of values,
  • as a graph in a coordinate system, and
  • in the form of a function equation.

Value tables are useful when individual value pairs are present, which is often the case, especially with empirically determined (measurement) data. With a large number of value pairs, however, tabular displays can quickly become confusing - without the use of computers. A second disadvantage is that missing function values ​​between two value pairs can only be estimated by averaging ("interpolation").

Representation of a functional relationship using a table of values.

In the case of graphic representations, the individual value pairs mapped in a unique way to points of a coordinate system. [3] If the distances between the value pairs are only very small, the functional relationship can be graphically illustrated by a curve. This often enables the function values ​​to be read off quickly (at least approximately). In this way, for example, the course of an electrical voltage signal over time can be observed directly on oscilloscopes or cardiograms. [4]

Representation of value pairs by means of a diagram (example: day length over the course of the year at the 50th degree of latitude).

How the picture of a function looks in a graphic representation also depends on the choice of the coordinate system, in particular on the scaling of the axes. For example, the - and the -Axis have different scalings, the function diagram appears distorted.

For the computational investigation of a function, the “analytical” form, ie a representation as a function equation, is preferred. A function equation can in turn be converted into a table of values ​​or a graphic form at any time, if required. A distinction is made between two types of functional equations:

  • In the explicit Form is the function equation after the (dependent) variable dissolved.


  • At a implicit Form kick the independent variable and the dependent variable on the same side of the equation; the equation thus has the form .


Not every function can be turned into a post resolved form are represented, for example . As far as possible, the explicit form of representation is generally used preferred. [5]

Surjectivity, injectivity and bijectivity¶

The distinction between surjective, injective and bijective functions enables an important classification of functions.

  • A function is called surjective if every element of its value set at least occurs once as a function value, i.e. each element of the value set is assigned to at least one element of the definition set.

    This property can be recognized on the diagram of a function by the fact that any, for -Axis parallel straight line intersects the function graph at least once in the entire value range.


  • A function is called bijective if every element of its set of values exactly occurs once as a function value, i.e. each element of the value set is assigned to exactly one element of the definition set. [6]

    This property can be recognized on the diagram of a function by the fact that any, for -Axis parallel straight line intersects the function graph exactly once in the entire value range.


    The function with the definition set and the set of values is bijective; the function graph is used by everyone -Axis of parallel straight lines intersected exactly once in the entire value range.

Every surjective or injective function can be made into a corresponding bijective function by appropriately restricting the set of definitions or values. [7]

Reversibility of a function¶

A function is a mathematical description of what "cause" a certain effect within a process evokes. Such a relationship only makes sense if the assignment of any value of the output variable to a result value is always unambiguous, a - So it's not worth two different ones Values ​​as a result.

Conversely, however, it is possible that different Values ​​the same - Deliver value as a result.


  • Different bodies can have the same mass. A single body, on the other hand, always has only one single, unambiguous value for the size of its mass.
  • In a fruit shop, a certain type of apple costs a definite price per quantity (at a certain point in time). Regardless of how many apples a customer actually buys, this clearly defines the total amount to be paid. The same price per quantity can, however, also apply to another type of fruit at the same time.

In general, functions are therefore not “reversible”, so it is not possible to find an assignment for every function that any one Value in a unique way Value assigns. A function has this property if and only if it is bijective. If a function is not bijective, it must first be made a bijective function by restricting its set of definitions or values.

Determination of the inverse function

The reverse function a function is found by looking at the original functional equation to resolves and then the variables and reversed.


If the inverse function of an inverse function is formed again according to the same principle, the original function is obtained again.

In the same coordinate system will be a function and their inverse function represented by the same function graph if only the naming of the - and Axis (argument and function values) can be exchanged. Should the names of the - and -Axis, on the other hand, remain, the graphs of a function and its inverse function are always axisymmetric to the bisector of the first and third quadrant.

Monotony and Narrowness¶

The investigation of a function for monotony, limitation, limit values ​​and continuity makes it possible in the field of analysis to make more far-reaching statements about the function, for example the appearance of the function graph.


In the same way as with sequences of numbers, monotony is an important characteristic of a function in functions.

Applies to all elements from the domain of a function, too , so the function is called monotonically increasing. Correspondingly, a function is called monotonically decreasing if for the function values ​​of all the condition applies. In the case of a constant function, the function values ​​are for all constant.

It therefore applies to every function and :

Applies to the above distinction instead of the less than or equal relation the smaller relation or instead of the greater than or equal to relation the greater than ratio , that's what the function is called strict monotonously decreasing or increasing. Every strictly monotonically increasing function is bijective and thus reversible; the inverse function has the same monotony as the original function.


One function is called bounded if there are two real numbers and there so that all function values lie between the two limiting numbers, so if:

Here is as the lower bound and referred to as the upper bound of the function.

A function can also only have a lower or an upper limit on one side in a certain area. For example, applies to all values ​​of the function the inequality so that any number represents an upper bound of the function. However, it is not possible to define a lower limit for the same function, since it assumes infinitely large values ​​in the negative range.

If a function has neither an upper nor a lower limit in a certain area, the function in this area is called unlimited.

Limits of a function¶

The values ​​of a function can - depending on the function type - just like the values ​​of a sequence of numbers with increasing -Values ​​approximate a certain numerical value. A function has such a limit if and only if it is monotonic and restricted.

Limits for and

Limits of functions are also defined in a very similar way to limits of sequences. However, while the “domain of definition” of sequences is limited to the natural numbers and thus only a Limit for can exist, can -Values ​​of functions in the positive as well as in the negative number range become infinitely large; it can therefore be a limit value for both as well as for define.

A limit of a function for exists precisely when there is ever greater Values ​​the associated -Values ​​always closer to a certain value approach. This is the case if and only for everyone Values ​​from a certain number the convergence criterion is met, i.e. the difference between becomes arbitrarily small. For every value, no matter how small must therefore apply:

This condition clearly states that you can use a "hose" as thin as you like (a so-called Environment) around the limit value can think around and then all function values ​​from a certain one Value must lie within this environment. [8]

Is there a limit value a function for any size negative or positive -Values, this is how you write:

Exists for a function one of the two limit values ​​above, then the function is called “convergent” for respectively . It is also possible that a function has no limit value for owns; in this case it is called divergent.


The function values ​​become a diverging function with increasing -Values ​​infinitely large, that's how they are called as an "improper" limit value - in fact there is no specific number in this case as the upper bound, as it actually has to exist for a limit value.

Limit for

Limit values ​​of functions can not only apply to infinitely large negative or positive Values ​​are considered; it is also possible to check whether a limit value exists if the Values ​​a freely selectable value approach. Does such a limit exist , so one writes:

Is the function at the point defined, its limit value at this point is equal to its function value, so it applies For . The above limit can, however, also exist if the function is in place is not defined. Especially at the limits of the domain of definition For this reason, functions are often examined for possible limit values ​​(for example at definition gaps).

If possible, approach the Values ​​of the body both from the left and from the right; one examines the behavior of the function at the points and , in which is as small a number as possible. So the following limit values ​​are formed:

The two associated limit values ​​are designated accordingly and as "left-sided" or "right-sided".

Calculation rules for limit values

There are the following calculation rules for calculating with limit values:

When dividing two functions or limit values, care must be taken not to divide by zero, so it has to be for all as be valid. Is in particular and a function with the limit value For , then:

Also applies to three functions and are the limit values the smallest and largest function are identical, this also applies to the limit value of the "average" function.


One describes a function in one place as continuous if the limit value on the left is at this point , the right-hand limit and the function value to match. A function is called (globally) continuous if the continuity condition for all -Values ​​of the definition range is fulfilled.

Clearly, continuity means that the graph of a function does not make any “jumps”, ie it can be drawn along the definition range as a solid line (without removing the pen). This is the case with a large number of functions, for example with all completely rational functions, the sine or cosine function. Also the tangent and hyperbolic function are continuous, as their function values ​​only change abruptly at the undefined points (definition gaps). The combination of two or more continuous functions using the basic arithmetic operations of addition, subtraction, multiplication or division other than zero results in a continuous function.

A clear example of a locally but not globally continuous function is the so-called signum function (also called the sign function). It is defined in sections as follows:

The Signum function is in all places except for (locally) continuously. At this point, however, your left-hand limit value is correct , their functional value and its right-hand limit do not match.

Intermediate value theorem and extreme value theorem

Is a function continuous in an interval, it is also limited there. So there is a lower bound and an upper bound , so that for all Values ​​of the interval applies.

Is a function in a closed interval continuous, the so-called extreme value theorem applies: In this case, two function values ​​can always be used and find so that applies. The value is taken as the minimum, the value as the maximum of the function in the interval designated.

One in a closed interval continuous function also takes every value between and at least once. This fact, which is particularly important for the numerical calculation of zeros, is called the "interim value theorem".


As a zero becomes an output value of a function, for which the associated function value takes the value zero:

The zeros of a function can be determined by looking into the implicit or explicit representation of the function for inserts the value zero and follows the resulting equation using algebraic methods dissolves. Depending on the type of function, it is possible that it has several, one or no zeros.

If a function is drawn in as a graph in a coordinate system, zeros represent points of intersection or contact with the -Axis.

Intersections of two functions

The determination of the interfaces of two or more functions is closely related to the determination of zeros. Consider two functions and so one can check for which one -Values ​​from the common domain the values ​​of the functions match for which output values so the condition applies. Solving this equation is formally consistent with determining the zero of match:

If one or more intersection points exist, the function values ​​of and usually not zero. You get the associated -Values ​​of the intersection points by taking those found when solving the above equation -Values ​​in one of the two functions.

Linking and chaining of functions¶

Further functions can be put together from the elementary functions, which are described in more detail in the next sections. This can be done in two ways:

  • In a so-called link, two functions are linked by one of the four basic arithmetic operations. The result of a function composed in this way is obtained by first calculating the values ​​of the two functions and then linking them with the corresponding basic arithmetic function.In this way, several functions can also be linked with one another step by step, paying attention to the order in which the links are evaluated (multiplication or division before addition or subtraction).

    In general, a linked function has the following form:

    Simple special cases of equation (5) arise here when one of the two functions is constant. This creates the following functions:

    In the first case, the constant becomes for every function value added (or subtracted, if is). In the case of a graphical representation, this changes the function graph Units shifted vertically (up for , down for ).

    In the second case, the function value is given with a constant multiplied. This will make the function graph in the case vertically compressed, in the case vertically stretched. Is , then the function graph (as with a centric stretching) is at the -Axis mirrored.

  • In a so-called chaining, two functions are executed “one after the other”, so the function value of the first function is used as the output value of the second function. This is generally only possible if the value range of the first function is a subset of the definition range of the second function.

    In general, a chained function has the following form:

    It will as outer and referred to as an inner function. Similar to the evaluation of terms in brackets, the first step is the value of the inner function calculated, and then, if allowed, as an argument for the outer function used.

When chaining two functions, the order in which they are chained must be observed: For example, is and , so is , while results.