Suppose that S is a locally compact separable metric space and C 0 is the set of continuous functions on S that vanish at infinity. To say a continuous function f vanishes at infinity means that given...

Suppose that S is a locally compact separable metric space and C₀
is the set of continuous functions on S that vanish at infinity. To say a continuous function f vanishes at infinity means that given ε > 0 there exists a compact set K such that
  if
  Show that if Assumption 20.1 is replaced by the assumptions that
  whenever f ∈ C₀
and
  uniformly as t → 0 whenever f ∈ C₀, then the conclusion of still holds.