Classes of continuously differentiable functions
From Maths
- 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.
Contents
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^{[1]}:
- Of class [ilmath]C^k[/ilmath] or
- [ilmath]k[/ilmath]-times differentiable
if [ilmath]f[/ilmath] has the following two properties:
- All partial derivatives of [ilmath]f[/ilmath] of order [ilmath]\le k[/ilmath] exist and
- All the partial derivatives are continuous on [ilmath]U[/ilmath]
Explicit cases
Class | Property | Meaning |
---|---|---|
[ilmath]C^0[/ilmath]^{[Note 1]} | 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]^{[Note 2]} we require^{[1]}:
- 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
- According to^{[1]} 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.
- Other content on page 645
See also
Notes
- ↑ This isn't really a special case, but it is worth stating explicitly
- ↑ This is inferred terminology, specifically John M. Lee uses the terms: "...functions of class [ilmath]C^k[/ilmath] on [ilmath]U[/ilmath] by..." when describing the set [ilmath]C^k(U)[/ilmath]
References
- ↑ ^{1.0} ^{1.1} ^{1.2} Introduction to Smooth Manifolds - John M. Lee - Second Edition - Springer GTM