# Smooth function

In mathematics, a**smooth function**is one that is infinitely differentiable, i.e., has derivatives of all finite orders. A function is called

**C**if it has a derivative that is a continuous function; such functions are also called

^{1}**continuously differentiable**. A function is called

**C**for

^{n}*n*≥ 1 if it can be differentiated

*n*times, with a continuous

*n*-th derivative. The smooth functions are therefore those that lie in the class C

^{n}for all

*n*. They are also called '''C

^{∞}functions.

For example, the exponential function is evidently smooth because the derivative of the exponential function is the exponential function itself.

Table of contents |

2 Relation to analytic function theory 3 Smooth partitions of unity 4 Smooth maps of manifolds |

## 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; so that Taylor's theorem cannot in general be applied to expand smooth functions.

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

- f(
*x*) = exp(-1/*x*),

*x*> 0. Not only do we have

- f(
*x*) → 0 as*x*→ 0 from above,

- P(
*x*)f(*x*) → 0

*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.

See also: An infinitely differentiable function that is not analytic

## Relation to analytic function theory

Thinking in terms of complex analysis, a function like

- g(
*z*) = exp(-1/*z*^{2})

*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.

## 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*.

*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 constrast, sheaves of smooth functions tend not to carry much topological information.

## 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.

See also: quasi-analytic function.