Smooth function

A smooth function is a mathematical function that is infinitely differentiable (in particular continuous). The term " smooth" is motivated by the intuition: The graph of a smooth function has no "corners", that is, places where it is not differentiable. Thus, the graph looks everywhere " particularly smooth ." For example, every holomorphic function is a smooth function. In addition, smooth functions are used as cutoff functions or test functions for distributions.

  • 3.1 Application

Definition

Conventions

For a non-empty open subset is called the set of real-valued and continuous functions on completely or with. Accordingly, the amount of natural once or a number is called the amount of times continuously differentiable functions with or with. The amount of times continuously differentiable function is recursively

Defined. It is always

Smooth functions

A function is called infinitely differentiable or smooth if for all. The set of all smooth functions on is listed with and it is

This description is particularly useful for topological considerations.

Generalizations

Without difficulty can the notion of smooth function generalized to more general cases. It is called a function is infinitely differentiable or smooth if all partial derivatives are infinitely differentiable. Also smooth functions are defined and examined between smooth manifolds.

Properties

  • Necessarily, all derivatives are continuous differentiable, since differentiability implies continuity.
  • Often found in mathematical considerations the term sufficiently smooth. By this is meant that the function is for a sufficiently large in, so just as often differentiable to perform the current train of thought. This is formulated in order to avoid a too strong (and not sensible ) restriction by " infinitely differentiable ", and not have to go through all the conditions on the other hand, are fulfilled in the cases usually considered anyway, or when the exact restriction for other reasons does not matter: As a theoretical argument can be argued that, for all are the times differentiable and the infinitely differentiable functions and the analytic functions with respect to many common metrics close to the continuous. If there is about a physical problem, in which small changes are not of importance, there is a continuous function arbitrarily considered " nearby" functions that meet the identified mathematical terms; possibly it can be shown even that proved the property for certain functions to a larger space in which they are close to, transfers. Is it clear from the context that only sufficiently smooth functions are considered ( eg by stating the degree of differentiability ) is "sufficiently " occasionally dispense with the addition.
  • In addition, one still called with the set of all analytic functions which are infinitely differentiable functions whose Taylor expansion around any point in an environment converges to the given function. It is worth noting, then, that each of the following inclusions in the real-valued case is genuine. In the case of complex-valued and complex differentiable, rather holomorphic functions on an open set any complex differentiable function is equal infinitely differentiable and even analytic. That is why the differentiability applies at most to functions whose definition and target quantity are the real numbers, vector spaces or manifolds over the real numbers or the like.
  • Each and also (as well ) is an ( infinite-dimensional ) vector space.

Examples

  • All polynomial functions are infinitely differentiable and even analytic.
  • The through defined function is twice continuously differentiable ( ), but the -th derivative is not differentiable at the point, ie.
  • The function is an infinitely differentiable function, but not an analytic function, because the Taylor series around the zero point is true in any environment around 0 with the function match because all the derivatives take the value 0 at 0.
  • Equally, however, is also infinitely differentiable. From local knowledge of an infinitely differentiable function so you can obviously no global statements derived (in this case applies, for example, all positive, but still ).
  • The Schwartz space contains only smooth functions and is a proper subset of infinitely differentiable functions.

Application

These last two examples are important tools for the construction of examples of smooth functions with special properties. Here is how you can a smooth partition of unity ( here: of ) construct:

  • Is infinitely differentiable with compact support.
  • Is infinitely differentiable and it holds

Topologization

Be an open subset. In the space of smooth functions a topology is explained in particular in the distribution theory. The family of semi-norms

With and through all compacta, makes the space of smooth functions on a locally convex space. This is completely and therefore a Fréchet space. In addition, since every closed and bounded set is compact, this is even a Montel space. The space of smooth functions with this locally -convex topology is usually labeled.

Bump function Topological space Closed set
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