Classes of continuously differentiable functions

Note: this page covers the use of [ilmath]C^k[/ilmath] on [ilmath]U[/ilmath], [ilmath]C^k(U)[/ilmath] for [ilmath]U\subseteq\mathbb{R}^n[/ilmath] and [ilmath]C^k(\mathbb{R}^n)[/ilmath] such.

Definition

Given [ilmath]U\subseteq\mathbb{R}^n[/ilmath] (where [ilmath]U[/ilmath] is open) and some [ilmath]k\ge 0[/ilmath], a function of the form:

• [ilmath]f:U\rightarrow\mathbb{R}^m[/ilmath] is given.

We may say [ilmath]f[/ilmath] is:

• Of class [ilmath]C^k[/ilmath] or
• [ilmath]k[/ilmath]-times differentiable

if [ilmath]f[/ilmath] has the following two properties:

1. All partial derivatives of [ilmath]f[/ilmath] of order [ilmath]\le k[/ilmath] exist and
2. All the partial derivatives are continuous on [ilmath]U[/ilmath]

Explicit cases

Class	Property	Meaning
[ilmath]C^0[/ilmath]	All continuous functions on [ilmath]U[/ilmath]	Here we take it as the class of functions whose zeroth-order partial derivatives exist and are continuous This is simply the function itself.
[ilmath]C^\infty[/ilmath]	Contains functions that are of class [ilmath]C^k[/ilmath] for all [ilmath]k\ge 0[/ilmath]	This is essentially a limit definition (see also smooth and diffeomorphism)

Notations

Of class [ilmath]C^k[/ilmath] on [ilmath]U[/ilmath]

To say a function is of class [ilmath]C^k[/ilmath] on [ilmath]U[/ilmath] we require:

• Here we must have [ilmath]U\mathop\subseteq_{\text{open} }\mathbb{R}^n[/ilmath] (with the standard topology (see Euclidean space))
• For an [ilmath]f[/ilmath] of class [ilmath]C^k[/ilmath] on [ilmath]U[/ilmath] we know:
  • All of the partial derivatives of [ilmath]f:U\rightarrow\mathbb{R}^m[/ilmath] (of order [ilmath]\le k[/ilmath]) exist and are continuous on [ilmath]U[/ilmath]
  • We do not know even what [ilmath]m[/ilmath] is.

Of class [ilmath]C^k(U)[/ilmath]

This is a set. It can be constructed and one can (sensibly) write [ilmath]f\in C^k(U)[/ilmath] (where as [ilmath]f\in C^k\text{ on }U[/ilmath] wouldn't be suitable and doesn't tell us the co-domain of [ilmath]f[/ilmath])

• Here [ilmath]U\subseteq\mathbb{R}^n[/ilmath] must be open (as before).
• By definition we denote:
  • The set of all real valued functions of class [ilmath]C^k[/ilmath] on [ilmath]U[/ilmath] as [ilmath]C^k(U)[/ilmath]
• All members are real-valued functions - these are functions with co-domain [ilmath]\mathbb{R} [/ilmath]

This means that for:

• A given open [ilmath]U\subseteq\mathbb{R}^n[/ilmath]

To say

• [ilmath]f\in C^k(U)[/ilmath] means that [ilmath]f:U\rightarrow\mathbb{R} [/ilmath] has continuous partial derivatives of all orders up to or equal to [ilmath]k[/ilmath] on [ilmath]U[/ilmath]

Unresolved issues

Warning: this section contains conflicts or ambiguities I am trying to resolve

1. According to smooth on [ilmath]A[/ilmath] works for an arbitrary [ilmath]A\subseteq\mathbb{R}^n[/ilmath] however:

   • He defines smooth as being of class [ilmath]C^k[/ilmath] for all [ilmath]k\ge 0[/ilmath]

   So to be smooth on [ilmath]A[/ilmath] is to be of class [ilmath]C^k[/ilmath] on [ilmath]A[/ilmath] forall [ilmath]k[/ilmath] - but we only define being of class [ilmath]C^k[/ilmath] for open subsets.

2. Other content on page 645

See also

• Index of notation