theorem
  for X0 being open non empty SubSpace of X, x being Point of X, x0
  being Point of X0 st x = x0 holds f is_continuous_at x iff f|X0
  is_continuous_at x0
proof
  let X0 be open non empty SubSpace of X, x be Point of X, x0 be Point of X0;
  assume
A1: x = x0;
  hence f is_continuous_at x implies f|X0 is_continuous_at x0 by Th58;
  thus f|X0 is_continuous_at x0 implies f is_continuous_at x
  proof
    reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
    assume
A2: f|X0 is_continuous_at x0;
    A is open by TSEP_1:16;
    hence thesis by A1,A2,Th60;
  end;
end;
