theorem Th60:
  for A being Subset of X, x being Point of X, x0 being Point of
  X0 st A is open & x in A & A c= the carrier of X0 & x = x0 holds f
  is_continuous_at x iff f|X0 is_continuous_at x0
proof
  let A be Subset of X, x be Point of X, x0 be Point of X0 such that
A1: A is open & x in A and
A2: A c= the carrier of X0 and
A3: x = x0;
  thus f is_continuous_at x implies f|X0 is_continuous_at x0 by A3,Th58;
  thus f|X0 is_continuous_at x0 implies f is_continuous_at x
  proof
    assume
A4: f|X0 is_continuous_at x0;
    A is a_neighborhood of x by A1,CONNSP_2:3;
    hence thesis by A2,A3,A4,Th59;
  end;
end;
