30. (a) Prove that if f x is continuous on a compact set KHX, where X; d...

30.

(a) Prove that if f ðxÞ is continuous on a compact set KHX, where ðX; dÞ is a metric space, then it is uniformly continuous on K. Assume that the range of f ðxÞ is a general metric space ðY; d 0Þ, or if easier, first consider the case where f : X ! R.

(Hint: First review the chapter proof when X ¼ R:Þ

(b) Show that if f ðxÞ ¼

Py j¼0 ajðx  x0Þ j is a power series that converges on I ¼ fx j jx  x0j < Rg;

and if fnðxÞ denotes the partial sum of this series, then fnðxÞ ! f ðxÞ uniformly on any compact set KHI .