Smooth function

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In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i.e., has derivatives of all finite orders:

A function is called C, or more commonly C⁰, if it is a continuous function.
A function is called C¹ if it has a derivative that is continuous; such functions are also called continuously differentiable.
A function is called Cⁿ for n ≥ 1 if it can be differentiated n times, leaving a continuous n-th derivative: such functions are also called finitely differentiable.
The smooth functions are those that lie in the class Cⁿ for all n; they are often referred to as C^∞ functions.

For example, the exponential function is evidently smooth because the derivative of the exponential function is the exponential function itself (which is continuous).

1 Constructing smooth functions to specifications
2 Topological vector spaces
3 Relation to analytic function theory
4 Smooth partitions of unity
5 Smooth maps of manifolds
6 Advanced definitions
7 See also

[edit] Constructing smooth functions to specifications

It is often useful to construct smooth functions that are zero outside a given interval, but not inside it. This is possible; on the other hand it is impossible that a power series can have that property. This shows that there is a large gap between smooth and analytic functions; and that in general smooth functions do not necessarily equal their Taylor series.

To give an explicit construction of such functions, we can start with a function such as

$f(x)=\left\{\begin{matrix} e^{-1/x^2} & \mbox{ for } x \neq 0 \\ 0 & \mbox{ for } x=0 \end{matrix}\right.$

Not only do we have

$\lim_{x \to 0} f(x) = 0$

we have

$\lim_{x \to 0} P(x)f(x) = 0$

for any polynomial P — because exponential growth with a negative exponent dominates. It follows that all derivatives of f(x) at zero are zero:

$f^{(n)}(0)=0 \mbox{ for all } n \,$

which means that setting f(x) = 0 for x ≤ 0 gives a smooth function. Combinations such as f(x)f(1-x) can then be made with any required interval as support; in this case the interval [0,1]. Such functions have an extremely slow 'lift-off' from 0.

[edit] Topological vector spaces

On a bounded domain D in Euclidean space, the set of functions C^k forms a Banach space with the norm

$\|f\|_k = \sup |f| + \sup |f'| + \cdots + \sup |f^{(k)}|$

however the set of smooth functions $\scriptstyle C^\infty$ is only a Frechet space.

[edit] Relation to analytic function theory

Thinking in terms of complex analysis, a function like

$g(z) = \exp\left(-\frac{1}{z^2}\right)$

is smooth for z taking real values, but has an essential singularity at z = 0. That is, the behaviour near z = 0 is bad; but it happens that one cannot see that, by looking at real arguments alone.

[edit] Smooth partitions of unity

Smooth functions with given closed support are used in the construction of smooth partitions of unity (see topology glossary for partition of unity); these are essential in the study of smooth manifolds, for example to show that Riemannian metrics can be defined globally starting from their local existence. A simple case is that of a bump function on the real line, that is, a smooth function f that takes the value 0 outside an interval [a,b] and such that

f(x) > 0 for a < x < b.

Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals (-∞, c] and [d,+∞) to cover the whole line, such that the sum of the functions is always 1.

From what has just been said, partitions of unity don't apply to holomorphic functions; their different behaviour relative to existence and analytic continuation is one of the roots of sheaf theory. In contrast, sheaves of smooth functions tend not to carry much topological information.

[edit] Smooth maps of manifolds

Smooth maps between smooth manifolds may be defined by means of charts, since the idea of smoothness of function is independent of the particular chart used. Such a map has a first derivative defined on tangent vectors; it gives a fibre-wise linear mapping on the level of tangent bundles.

[edit] Advanced definitions

When one needs to talk about the set of all infinitely differentiable functions, and how elements of that space behave when differentiated and integrated, summed and taken limits of, then it turns out that the space of all smooth functions is an inappropriate choice, as it fails to be complete and closed under these operations. For a proper treatment in this case, the concept of a Sobolev space must be used.