Abstract
We consider the space C^1(K) of real-valued continuously differentiable functions on a compact set Ksubseteq mathbb {R}^d. We characterize the completeness of this space and prove that the restriction space C^1(mathbb {R}^d|K)={f|_K: fin C^1(mathbb {R}^d)} is always dense in C^1(K). The space C^1(K) is then compared with other spaces of differentiable functions on compact sets.
Highlights
In most analysis textbooks differentiability is only treated for functions on open domains and, if needed, e.g., for the divergence theorem, an ad hoc generalization for functions on compact sets is given
We will show that the space C1(Rd|K) of restrictions of continuously differentiable functions on Rd to K is always dense in C1(K)
This implies that y is the endpoint of gap G = (a, y) with a ≥ x which implies
Summary
In most analysis textbooks differentiability is only treated for functions on open domains and, if needed, e.g., for the divergence theorem, an ad hoc generalization for functions on compact sets is given. We propose instead to define differentiability on arbitrary sets as the usual affine-linear approximability—. We will only consider compact domains in order to have a natural norm on our space. The results are extended to σ-compact (and, in particular, closed) sets. An Rn-valued function f on a compact set K ⊆ Rd is said to belong. C1(K, Rn) if there exits a continuous function df on K with values in the linear maps from Rd to Rn such that, for all x ∈ K, f (y) − f (x) − df (x)(y − x) yli→mx
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
