Differentiable function

As differentiability is called in mathematics the property of a function to be locally around a point in an unambiguous manner approximating linear. The concept of differentiability is not only for real-valued functions declared on the set of real numbers, but also for functions of several variables, for complex functions, for mappings between vector spaces and for many other types of functions and images. For some types of functions (for example, for functions of several variables) there are several different Differenzierbarkeitsbegriffe.

The question of the differentiability is one of the problems of differential calculus, a section of the Analysis.

• 3.1 Partial differentiability
• 3.2 directional derivative
• 3.3 Total differentiability
• 3.4 interrelationships between the various Differenzierbarkeitsbegriffen
• 3.5 Examples
• 3.6 counterexamples 3.6.1 Partially differentiable but not continuous and not all directional derivatives
• 3.6.2 Unilateral, but not bilateral directional derivatives
• 3.6.3 All directional derivatives exist, but do not define a linear map
• 3.6.4 All directional derivatives exist and define a linear map, but not totally differentiable
• 3.6.5 Total differentiable but not continuously differentiable

• 5.1 Gâteaux - differentiability
• 5.2 Fréchet differentiability
• 5.3 relationships

Real-valued functions of a real variable

Definitions

In the simplest case, we consider a real-valued function of a real variable, that is, a function whose function values ​​are real numbers, and the domain is an open interval of real numbers. Such a function is differentiable at a point of its domain, when the derivative of exists at this point. There are essentially two equivalents of Definition of the existence of the derivative.

Both definitions are equivalent: Is differentiable with respect to the first definition, one chooses for the limit from the definition and uses

Then the first property on the choice of and satisfied and the second because

Conversely, if the second definition are met, is obtained by forming the first property

And the limit of the difference quotient exists due

If a function is differentiable at a point, to write for the derivation.

A function is said to be differentiable (without limitation to a specific point) if it is differentiable at every point of its domain. The function is called the derivative function or short derivation of.

Notes

Graphically, can the property differentiability be interpreted that a function if and only on the site is differentiable if the corresponding point on the graph exists of exactly one tangent that is not perpendicular. The tangent is the graph referred to in the second definition of linear function.

The derivative at the point of the gradient of this tangent. The difference quotients referred to in the first definition, the slopes of secants through the point and another point on the curve. So the function is differentiable at the point when the slopes of these secants converge for the border crossing against against the slope of the tangent.

From continuity differentiability follows: Each differentiable at a point function is also continuous there. Every differentiable on their domain function is continuous. The converse is not true. The below is not differentiable functions are all continuous.

Examples of differentiable functions

From the derivation rules follows:

• Any function that can be represented by a polynomial is differentiable.
• Sums, products and quotients of differentiable functions are differentiable.
• Concatenations of differentiable functions are differentiable.
• The inverse function a bijective differentiable function is exactly then at the point differentiable if is.

From the limit theorems for power series follows:

• Any function that can be represented locally by a power series is differentiable.

Examples of non- or not everywhere differentiable functions

Since each function is continuously differentiable, is reversed every non- continuous function (for example, a step function or Dirichlet function) is an example of a non- differentiable function. However, there are other functions that are continuous, but not, or not differentiable everywhere.

Root function

The root function is not differentiable at the point. The difference quotient

Strives for infinity, so does not converge. The graph of the function has a tangent at the point, but it extends vertically and therefore has no slope.

Absolute value function

The absolute value function is not differentiable at the point 0.

For is and thus

For on the other hand, and consequently

Since the left - and right-sided limits do not match, the limit does not exist. The function is therefore not differentiable at the point considered.

There are at the point 0, however, the right -hand derivative

And the left derivative

The function graph has a kink at the point 0. There is sort of a left-sided tangent with slope -1 and a right-sided with slope 1. There is a line for each slope between -1 and 1, the function of the graph at the point ( 0,0) "touched", but not " hugs ".

This is a typical behavior for partially defined functions, where at the seams while match the function values ​​, but not the derivatives. The graphs of differentiable functions in contrast, have no kinks.

A third example

The function

Is continuous at 0 but not differentiable (but everywhere else ). For the difference quotient at the point 0

The limit for does not exist. There are no one-sided limits. Rather, the difference quotient commutes when goes to 0, infinitely often between the values ​​-1 and 1 and takes every intermediate value infinitely often in.

Weierstrass function

Named after its discoverer Weierstrass function

Is everywhere continuous but nowhere differentiable.

Wiener process

Other examples provides the mathematical Brownian motion: Almost every path of a Wiener process is continuous as a function, but nowhere differentiable.

Continuous differentiability and higher derivatives

A function is called continuously differentiable if it is differentiable and its derivative is continuous. Even if the function is everywhere differentiable, the derivative does not have to be continuous. For example, the function

At every point, including, differentiable. The derivation

But is not continuous at 0.

Because of the differentiability of a function followed by the continuity in a twice- differentiable function of the function itself and the first derivative are automatically continuously. However, the second discharge does not need to be continuous. According to the function itself and all leads, ... up to the - th derivative are continuous at - times differentiable function. For the -th derivative, however, this need not apply. Is this also continuous, then you called times continuously differentiable. Are all leads back differentiable, then the function is called infinitely differentiable or smooth.

The set of all - times continuously differentiable functions with the definition of quantity is called. The set of infinitely differentiable functions is called. A - times continuously differentiable function are therefore also called a function of Differentiationsklasse in short, function of the class or function. An infinitely differentiable function is called according to the function ( differentiation ) class or function.

The function

Is differentiable, its derivative is the function that is continuous but not differentiable at the point 0. The function is thus continuously differentiable, but at the point 0 is not twice differentiable. Accordingly, the function

Times continuously differentiable, but not at the point 0 times differentiable.

Complex Functions

For complex functions, ie complex-valued functions of a complex variable, we define differentiability analogous to real functions. It should be an open subset of the complex plane and a point of this subset. A function is called complex differentiable at the point, if the limit

Exists. In this case, this is referred to as a limit.

A function is called holomorphic in point, if a neighborhood exists, in which is complex differentiable. Holomorphic functions are automatically infinitely complex differentiable and even analytic.

Real-valued functions of several variables

For functions of several variables, ie functions that are defined on open subsets of Euclidean space, there are several different concepts of strong differentiability. Below is an open set. The elements of a tuple can be written. Next a function is given. We consider a fixed point and consider differentiable at the point.

Partial differentiability

This is the weakest Differenzierbarkeitsbegriff. The function is called differentiable at the point in direction if the partial derivative

Exists. So you considered all variables to be constant and to consider the resulting function of one variable.

The function is called differentiable if at any point, all partial derivatives exist. It is continuous differentiable if all partial derivatives are continuous functions from to.

From partial differentiability does not follow the continuity, but continuity in the direction of the coordinate axes.

Directional derivative

Is a unit vector, the ( two-sided ) directional derivative in the direction of is defined at the point when

Considering only positive, we obtain the one-sided directional derivative

The function is called ( one-sided) differentiable in the direction of if the (one-sided ) directional derivative in the direction of exist. The directional derivatives in the direction of the unit vectors of the standard basis are just the partial derivatives

Total differentiability

The function is called totally differentiable at the point, if a linear mapping and a function exist, so that makes up for the error by approximating,

And of higher than first order goes to 0, that is for.

The linear mapping is called total derivative of the point. It is called with. The matrix representation with respect to the standard basis is the Jacobian matrix and is designated by or. The function is called totally differentiable if it is totally differentiable at each point.

A totally differentiable function is also continuous.

Relationships between the various Differenzierbarkeitsbegriffen

• If both sides differentiable in any direction, so is particularly partial differentiable.
• Is totally differentiable, then is differentiable in each direction ( ie, in particular also differentiable ). The entries in the Jacobian matrix are the partial derivatives with.

The reversals are not applicable:

• From the partial differentiability follows either the total or the two-sided or one-sided differentiability differentiability in directions which are not co-ordinate directions.
• Also from the two-sided differentiability in all directions does not follow total differentiability. Not even if the candidate for the total derivative, the picture is linear.

The situation is different if one presupposes not only the existence but also the continuity of the partial derivatives.

• Is continuously differentiable, so is totally differentiable.

They call continuously partial differentiable functions therefore simply continuously differentiable. Again, the converse is not true:

• From total differentiability does not follow the continuity of the partial derivatives.

Overall, therefore, the following applies:

However, it is none of the reversals.

Examples

• Any function that can be represented as a polynomial in the variables is continuously differentiable.
• Sums, products, quotients and concatenations of continuously differentiable functions are continuously differentiable.

Counter-examples

All counterexamples are features on the. The coordinates are denoted with and instead of and. Of interest here is only the differentiability and continuity at the origin. Everywhere else the functions are continuously differentiable.

Differentiable but not continuous and not all directional derivatives

The function

Is differentiable at the point (0,0). On the coordinate axes, the function has a constant value of 0, that is, for all, and is

It follows

However, the function is not continuously at (0,0 ). On the first bisector (except for the origin) has a constant value of one (). One approaches the origin on the first bisector, the function values ​​approach so against 1 The directional derivative in directions other than that of the coordinate axes do not exist.

The function

Is differentiable and continuous at the location (0,0). All one-sided directional derivatives exist, but except in the coordinate directions not two-sided.

Unilateral, but not bilateral directional derivatives

The Euclidean norm

Generalizes the absolute value function. It is continuous everywhere.

For each unit vector of the one-sided directional derivative exists in and it is

However, the two-sided directional derivatives do not exist, because otherwise would apply. In particular, the function is not differentiable.

All directional derivatives exist, but do not define a linear map

Here there are all directional derivatives, applies to the partial derivatives

However, the picture is not linear. For the unit vector

During

All directional derivatives exist and define a linear map, but not totally differentiable

Here there are all directional derivatives, applies to each vector. In particular, is differentiable with

And Figure

Is the zero mapping, so trivially linear.

The function is also continuous. However, it is not totally differentiable at the point (0,0). If it were, it would be the zero mapping and for each vector would apply

For the error term so would apply

Putting and, we obtain

For to 0, this term goes to 0 instead of against

Total differentiable but not continuously differentiable

This function is the corresponding sample function of a variable modeled, the proof proceeds in principle the same as there.

The function is totally differentiated on the body (0,0), the derivation of the null function. However one approaches the zero point, diverge the partial derivatives, for example, is the amount of

Tends to infinity for 0

Maps between finite-dimensional vector spaces

An illustration of an open set in the vector space can be represented by its component functions:

Differentiability of can then be traced back to the differentiability. is ( at the point ) if and only partially differentiable ( differentiable in the direction of the vector, totally differentiable, continuously differentiable ) if all component functions have this property.

Is the point totally differentiable, then by a linear mapping of. Her performance matrix, the Jacobian matrix consists of the partial derivatives

And the direction of the discharge point in the direction of the image is of the vector of the linear mapping.

Functions and images on infinite-dimensional vector spaces

In infinite-dimensional vector spaces, there are no coordinates, so there is no partial differentiability. The terms directional derivative and total differentiability can be generalized, however, to infinite -dimensional vector spaces. In this case, in contrast to the finite plays the topology to the vector spaces play an important role. Typical example of infinite-dimensional vector spaces are function spaces, ie vector spaces whose " vectors" are functions. To distinguish called defined on these vector spaces functions Functional and called mappings between vector spaces such operators.

Gâteaux - differentiability

→ Main article: Gâteaux differential

The directional derivative corresponding to the Gâteaux derivative. Given a normed vector space (ie, a (typically infinite-dimensional ) vector space together with a norm ), an open subset and a functional. The Gâteaux derivative of at a "point" in the direction of a vector is then given by

If the limit exists.

If the Gâteaux - derivative exists for each, then a figure explains. From the definition it follows immediately that this figure is positively homogeneous, ie for all. As in the finite-dimensional follows from the existence of all directional derivatives not think that is additive and thus linear. Although the mapping is linear, it does not follow that it is continuous.

For the notion of Gâteaux - differentiability, there are several non-compatible conventions:

Some authors call a functional Gâteaux - differentiable at the point, if any exist, and then describe the picture as Gateaux derivative of the point. Others require in addition that is linear and continuous.

Analogously one defines Gâteaux - differentiability and Gâteaux - derivative for operators from a normed vector space into another normed vector space (typically a Banach space ). The requirement in the definition of the Gâteaux derivative convergence then be understood in terms of the standard of. The same applies to the continuity of.

Fréchet differentiability

→ Main article: Fréchet derivative

The total differentiability in the finite corresponds to the Fréchet differentiability in infinite-dimensional vector spaces. Given Banach spaces and an open subset, an image and a point.

The mapping is called Fréchet differentiable if a limited (ie, continuous) linear map and a mapping exists, such that for all with

And

It is in the numerator of the norm, in the denominator of the.

The linear operator in this case means the Fréchet derivative of at the point.

Relationships

As in the finite every Fréchet differentiable map is Gâteaux - differentiable and the Gâteaux - derivative coincides with the Fréchet derivative agreement. Conversely needs at the point even then not to be Fréchet differentiable if the Gâteaux - derivative is linear and continuous.

Differentiable mappings between differentiable manifolds

The differentiability of maps between differentiable manifolds is attributed to the differentiability of their map representations. This continuity must already be presupposed.

Let and be differentiable manifolds of dimensions, respectively, and the Differenzierbarkeitsklasse and it is a continuous map. Then there exists for every point a map of order, that is an open environment that includes, and a homeomorphism defined on open to a subset of the. The same is also a map of around the pixel. Since is continuous, the map can be chosen so that all in lies. Under the map display of respect to these maps are then refers to mapping

This is a map of the open portion of the open portion in the amount of.

The mapping is called continuously differentiable if it is continuous and its map representations are continuously differentiable. They say times continuously differentiable ( for ), or from the class if their maps of - times continuously differentiable.

The differentiability does not depend on the choice of maps from (as long as is ), since the map change pictures - diffeomorphisms are. Is or Euclidean space, so there you can dispense with the card. In particular:

A function is iff - times continuously differentiable if the cards for their representations regarding cards applies.

Analogously one defines the complex differentiability for complex-valued functions on complex manifolds and mappings between complex manifolds.

For the definition of the derivative of a map between manifolds or function on a manifold see tangent and push forward.

Term extensions

The following concepts are generalizations of differentiability:

• Weak derivatives
• Differentiability in the sense of distributions
• Radon Nikodým derivation