  We prove two theorems about differentiable functions on the Banach space
C(K), where K is compact.
  (i) If C(K) admits a non-trivial function of class C^m and of bounded
support, then all continuous real-valued functions on C(K) may be uniformly
approximated by functions of class C^m.
  (ii) If C(K) admits an equivalent norm with locally uniformly convex dual
norm, then C(K) admits an equivalent norm which is of class C^infty (except at
0).
