
theorem
  for X be non empty TopSpace
  for x being set st x in C_0_Functions(X) holds
  x in ContinuousFunctions(X)
proof
  let X be non empty TopSpace;
  let x be set such that
A1:  x in C_0_Functions(X);
   consider f be RealMap of X such that
A2: f=x & f is continuous
    & (ex Y be non empty Subset of X st Y is compact
    & (for A being Subset of X st A=support(f)
    holds Cl(A) is Subset of Y)) by A1;
  thus thesis by A2;
end;
