theorem Th44:
  f is continuous iff for x being Point of X holds f is_continuous_at x
proof
  thus f is continuous implies for x being Point of X holds f is_continuous_at
  x;
  assume
A1: for x being Point of X holds f is_continuous_at x;
  thus f is continuous
  proof
    let x be Point of X, G be a_neighborhood of f.x;
    f is_continuous_at x by A1;
    hence thesis;
  end;
end;
