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In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of any order are called smooth. Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ...
Partial plot of a function f. ...
In some places this article assumes an acquaintance with algebra, analytic geometry, or the limit. ...
Most of the discussion in this article will be about real-valued functions of one real variable. A discussion of the multivariable case will be presented towards the end. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
Differentiability classes Consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer. The function f is said to be of class Ck if the derivatives f', f'', ..., f(k) exist and are continuous (the continuity is automatic for all the derivatives except the last one, f(k)). The function f is said to be of class C∞, or smooth, if it has derivatives of all orders. f is said to be of class Cω, or analytic, if f is smooth and if it equals its Taylor series expansion around any point in its domain. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
In mathematics, the real line is simply the set of real numbers. ...
The integers are commonly denoted by the above symbol. ...
In mathematics, an analytic function is a function that is locally given by a convergent power series. ...
As the degree of the Taylor series rises, it approaches the correct function. ...
To put it differently, the class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous. Thus, a C1 function is exactly a function whose derivative exists and is of class C0. In general, the classes Ck can be defined recursively by declaring C0 to be the set of all continuous functions and declaring Ck for any positive integer k to be the set of all differentiable functions whose derivative is in Ck-1. In particular, Ck is contained in Ck-1 for every k, and there are examples to show that this containment is strict. C∞ is the intersection of the sets Ck as k varies over the non-negative integers. Cω is strictly contained in C∞; for an example of this, see bump function or also below. A visual form of recursion known as the Droste effect. ...
In mathematics, a smooth function is one that is infinitely differentiable, i. ...
Examples
 The function f( x)= x for x≥0 and 0 otherwise.
 The function f( x)= x2 sin(1/ x) for x>0.
 A smooth function that is not analytic. The function Image File history File links Size of this preview: 800 × 534 pixelsFull resolution (1153 × 769 pixel, file size: 9 KB, MIME type: image/png) function main() % set up the plotting window thick_line=4; thin_line=2; arrow_size=4; arrow_type=2; fs=30; circrad=0. ...
Image File history File links Size of this preview: 800 × 534 pixelsFull resolution (1153 × 769 pixel, file size: 9 KB, MIME type: image/png) function main() % set up the plotting window thick_line=4; thin_line=2; arrow_size=4; arrow_type=2; fs=30; circrad=0. ...
Image File history File links Size of this preview: 732 × 600 pixelsFull resolution (786 × 644 pixel, file size: 11 KB, MIME type: image/png) function discontinuity() % set up the plotting window thick_line=2. ...
Image File history File links Size of this preview: 732 × 600 pixelsFull resolution (786 × 644 pixel, file size: 11 KB, MIME type: image/png) function discontinuity() % set up the plotting window thick_line=2. ...
Image File history File links Mollifier_illustration. ...
Image File history File links Mollifier_illustration. ...
 is continuous, but not differentiable, so it is of class C0 but not of class C1.
The function
 is differentiable, with derivative  Because cos 1/x oscillates as x approaches zero, f '(x) is not continuous at zero. Therefore, this function is differentiable but not of class C1.
The function
 is smooth, so of class C∞, but it is not analytic, so it is not of class Cω.
The exponential function is analytic, so, of class Cω. The trigonometric functions are also analytic wherever they are defined. The exponential function is one of the most important functions in mathematics. ...
All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other...
The space of Ck functions Let D be an open subset of the real line. The set of all Ck functions defined on D and taking real values is a Fréchet space with the countable family of seminorms This article deals with Fréchet spaces in functional analysis. ...
In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of length. A space with such a seminorm is then known as a seminormed space. ...
 where K varies over an increasing sequence of compact sets whose union is D, and m=0, 1, …, k. In mathematics, a sequence is an ordered list of objects (or events). ...
In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ...
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
The set of C∞ functions over D also forms a Fréchet space. One uses the same seminorms as above, except that m is allowed to range over all non-negative integer values.
In many applications where functions having derivatives of certain order are necessary, the above spaces turn out to be not the most convenient. One often uses Sobolev spaces instead, which, roughly speaking, consist of functions for which the k-th derivative is not differentiable, but rather, integrable. In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its weak derivatives up to some order k the condition of finite Lp norm, for given p ≥ 1. ...
Differentiability classes in several variables Let n and m be some positive integers. If f is a function from an open subset of Rn with values in Rm, then f has component functions f1, ..., fm. Each of these may or may not have partial derivatives. We say that f is of class Ck if all of the partial derivatives  exist and are continuous, where each of  is an integer between 1 and n. The classes C∞ and Cω are defined as before. In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...
These criteria of differentiability can be applied to the transition functions of a differential structure. The resulting space is called a Ck manifold. In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes it into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
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 In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...
This is a glossary of some terms used in the branch of mathematics known as topology. ...
In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ...
In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...
In mathematics, a smooth function is one that is infinitely differentiable, i. ...
- 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 behavior 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. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ...
In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...
See also - Non-analytic smooth function
- Quasi-analytic function
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