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In mathematics, a smooth function is one that is
infinitely differentiable, i.e., has derivatives of all finite orders. A
function is called C1 if it has a derivative that is a continuous function; such functions are also called continuously differentiable. A
function is called Cn for 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 Cn
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.
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),
defined initially for x > 0. Not only do we have
- f(x) → 0 as x → 0 from above,
we have
- P(x)f(x) → 0
for any polynomial P — because exponential growth with a negative exponent dominates. That 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.
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/z2)
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.
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 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.
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