Continuously Differentiable Functions on Compact Sets
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Continuously Differentiable Functions on Compact Sets Leonhard Frerick, Laurent Loosveldt, and Jochen Wengenroth Abstract. We consider the space C 1 (K) of real-valued continuously differentiable functions on a compact set K ⊆ Rd . We characterize the completeness of this space and prove that the restriction space C 1 (Rd |K) = {f |K : f ∈ C 1 (Rd )} is always dense in C 1 (K). The space C 1 (K) is then compared with other spaces of differentiable functions on compact sets. Mathematics Subject Classification. 46E10, 46E15, 26B35, 28B05. Keywords. Differentiability on compact sets, Whitney jets.
1. Introduction 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— the price one has to pay is then the definite article: Instead of the derivative there can be many. We will only consider compact domains in order to have a natural norm on our space. The results are easily extended to σ-compact (and, in particular, closed) sets. An Rn -valued function f on a compact set K ⊆ Rd is said to belong 1 C (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, lim
y→x y∈K
f (y) − f (x) − df (x)(y − x) = 0, |y − x| 0123456789().: V,-vol
(1)
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where | · | is the euclidean norm. For n = 1 we often identify Rd with its dual and write ·, · for the evaluation which is then the scalar product. Questions about C 1 (K, Rn ) easily reduce to the case C 1 (K) = C 1 (K, R). Of course, equality (1) means that df is a continuous (Fr´echet) derivative of f on K. As in the case of open domains, every f ∈ C 1 (K) is continuous and we have the chain rule: For all (continuous) derivatives df of f on K and dg of g on f (K) the map x → dg(f (x)) ◦ df (x) is a (continuous) derivative of g ◦ f on K. In general, a derivative need not be unique. For this reason, a good tool to study C 1 (K) is the jet space J 1 (K) = {(f, df ) : df is a continuous derivative of f on K} endowed with the norm (f, df ) J 1 (K) = f K + df K , where · K is the uniform norm on K and |df (x)| = sup{|df (x)(v)| : |v| ≤ 1}. For the projection π(f, df ) = f we have C 1 (K) = π(J 1 (K)), and we equip C 1 (K) with the quotient norm, i.e., f C 1 (K) = f K + inf{ df K : df is a continuous derivative of f on K}. It seems that the space C 1 (K) did not get much attention in the literature. This is in sharp contrast to the “restriction space” C 1 (Rd |K) = {f |K : f ∈ C 1 (Rd )}. Obviously, the inclusion C 1 (Rd |K) ⊆ C 1 (K) holds but it is wellknown that, in general, it is strict. Simple examples are domains with inward directed cusps like K = {(x, y) ∈ [−1, 1]2 : |y| ≥ e−1/x for x > 0}. The function f (x, y) = e−1/(2x) for x, y > 0 and f (x, y) = 0 elsewhere, is in C 1 (K) but it is not t
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