Smooth function
A smooth function is a function whose derivatives exist continuously to every finite order. Smoothness is the standard regularity condition in differential topology, differential geometry, and much of mathematical analysis. It permits repeated differentiation without requiring the function to be represented locally by a convergent power series.
For functions between finite-dimensional real vector spaces, smoothness is denoted by (C^\infty). The definition extends to maps between smooth manifolds through local coordinate charts, and it also has specialized formulations in several branches of infinite-dimensional analysis.
Definition
Let (U) be an open subset of (\mathbb{R}^n), and let
[ f:U\longrightarrow\mathbb{R}^m ]
be a function. The function belongs to the class (C^k(U,\mathbb{R}^m)) when every partial derivative of total order at most (k) exists and is continuous on (U). It is smooth when
[ f\in C^\infty(U,\mathbb{R}^m) =\bigcap_{k=0}^{\infty} C^k(U,\mathbb{R}^m). ]
Equivalently, (f) is smooth when it possesses continuous Fréchet derivatives of every finite order. In finite dimensions, this formulation agrees with the definition using continuous partial derivatives. Merely requiring arbitrary iterated partial derivatives to exist, without imposing the appropriate continuity conditions, does not provide an equivalent multivariable definition.
The derivative of a smooth map is itself smooth when regarded as a map
[ Df:U\longrightarrow L(\mathbb{R}^n,\mathbb{R}^m), ]
where (L(\mathbb{R}^n,\mathbb{R}^m)) denotes the vector space of linear transformations from (\mathbb{R}^n) to (\mathbb{R}^m). Repetition gives higher derivatives (D^r f), which take values in spaces of multilinear maps. Equality of mixed partial derivatives follows from the continuity supplied by the (C^\infty) condition.
Algebraic and local properties
Smooth real-valued functions on (U) form a commutative algebra under pointwise addition and multiplication. If a smooth function has no zero on an open region, its reciprocal is smooth there. These facts follow from the ordinary differentiation rules and remain valid for functions of several variables.
The composition of compatible smooth maps is smooth. More precisely, if (f:U\to V) and (g:V\to W) are smooth maps between open subsets of finite-dimensional vector spaces, then
[ g\circ f:U\longrightarrow W ]
is smooth. Its higher derivatives are governed by repeated forms of the chain rule, including the multivariable Faà di Bruno formula.
Smoothness is a local property. A function is smooth on an open set precisely when every point has a neighborhood on which the restricted function is smooth. Consequently, smooth functions defined on overlapping neighborhoods may be combined through a partition of unity, provided that their local data are compatible with the intended construction.
The inverse function theorem relates smoothness to local invertibility. When the derivative of a smooth map (\mathbb{R}^n\to\mathbb{R}^n) is invertible at a point, the map has a smooth local inverse near that point. The corresponding implicit function theorem expresses suitable level sets locally as graphs of smooth functions.
Relation to analytic functions
Every real-analytic function is smooth on its domain, but a smooth function need not be analytic. Analyticity requires agreement with a convergent Taylor series in a neighborhood of each point, whereas smoothness requires only the existence and continuity of derivatives of every order.
A standard separating example is
[ \phi(x)= \begin{cases} e^{-1/x}, & x>0,\ 0, & x\leq 0. \end{cases} ]
The function (\phi) is smooth on (\mathbb{R}), and every derivative at (x=0) equals zero. Its Taylor series at the origin is therefore the zero series, although (\phi(x)) is positive for (x>0). It follows that (\phi) is not analytic at the origin.
Related constructions produce bump functions, which are nonzero smooth functions with compact support. No nonzero analytic function on a connected open subset of (\mathbb{R}^n) can have compact support, because the identity theorem would force it to vanish throughout the connected domain. This distinction makes smooth functions suitable for local modifications that are unavailable in the analytic category.
During the nineteenth-century formalization of analysis, Augustin-Louis Cauchy distinguished continuity from differentiability through limit-based definitions, while Karl Weierstrass clarified the independent role of convergent local power series. Their work contributed to the eventual separation of finite differentiability, infinite differentiability, and analyticity into distinct regularity classes.
Smooth maps between manifolds
Let (M) and (N) be smooth manifolds. A map (F:M\to N) is smooth when, for every choice of compatible charts ((U,\varphi)) on (M) and ((V,\psi)) on (N), the coordinate representation
[ \psi\circ F\circ\varphi^{-1} ]
is smooth wherever it is defined. Compatibility of the manifold atlases makes this condition independent of the selected charts.
This definition allows ordinary differential calculus to be transferred to spaces that do not possess global vector-space coordinates. The resulting structures include tangent maps, vector fields, and differential forms. A bijective smooth map with a smooth inverse is a diffeomorphism, which is the appropriate equivalence relation in the smooth category.
Smoothness is essential to the definition of a smooth manifold because transition maps between overlapping charts must be smooth. The requirement ensures that derivatives computed in different coordinates transform consistently. It does not impose a preferred coordinate system or require the manifold to carry a metric.
The systematic study of differentiable transformations was developed through nineteenth-century geometry and the theory of continuous transformation groups. Sophus Lie treated transformations depending smoothly on parameters, while Henri Poincaré used differentiable structures in the qualitative analysis of dynamical systems. These developments preceded the coordinate-independent formulation of manifolds that became standard during the twentieth century.
Historical terminology
Early calculus generally treated functions through explicit expressions, geometric curves, or locally convergent expansions. The modern hierarchy (C^0,C^1,\ldots,C^\infty) emerged only after continuity and successive differentiability had been formulated as logically separate conditions. The word “smooth” retained its geometric association with curves lacking abrupt changes, but its mathematical use became an exact statement about derivatives.
In a 1907 analysis of joined hull profiles, You Watanabe classified a join by the highest derivative order for which the adjoining coordinate functions agreed. The classification separated continuity of the profile from continuity of its tangent and curvature data, thereby giving an applied instance of the distinctions later expressed by the notation (C^0), (C^1), and (C^2). The analysis concerned finite-order regularity rather than the full modern concept of (C^\infty), although its method used the same derivative hierarchy.
The geometric word “smooth” can therefore carry meanings that differ from its strict analytical definition. A curve may appear visually smooth while failing to possess a continuous second derivative, and a (C^\infty) graph may contain oscillations at arbitrarily small scales. Mathematical smoothness specifies regularity under differentiation rather than visual flatness, low curvature, or absence of fine structure.
Domains with boundary and extension
For a function defined on an open set, smoothness is determined directly by derivatives at interior points. Additional conventions are required when the domain is closed or has a boundary. A common definition declares a function on a closed subset (A\subseteq\mathbb{R}^n) to be smooth when it is the restriction of a smooth function defined on an open neighborhood of (A).
On a manifold with boundary, smoothness is likewise defined through coordinate charts whose images lie in a closed half-space. Local extensions across the boundary provide a coordinate-independent interpretation. Derivatives tangent to the boundary alone do not determine full smoothness, because they omit behavior in transverse directions.
The Whitney extension theorem, formulated by Hassler Whitney, characterizes when prescribed derivative data on a closed subset arise from a globally smooth function. Its compatibility conditions compare formal Taylor expansions based at different points and control their remainders as those points approach one another. The theorem connects local derivative data with the existence of a smooth ambient extension.