
theorem
for f be PartFunc of REAL,REAL, x0 be Real st
 f is_convergent_in x0 & lim(f,x0) = f.x0 holds f is_continuous_in x0
proof
    let f be PartFunc of REAL,REAL, x0 be Real;
    assume that
A1:  f is_convergent_in x0 and
A2:  lim(f,x0) = f.x0;

    for e be Real st 0 < e ex d be Real st 0 < d &
     for x be Real st x in dom f & |. x-x0 .| < d holds |. f.x-f.x0 .|< e
    proof
     let e be Real;
     assume
A3:   0 < e; then
     consider d be Real such that
A4:   0 < d &
      for x be Real st 0 < |. x0-x .| & |. x0-x .| < d & x in dom f
        holds |. f.x-f.x0 .| < e by A1,A2,LIMFUNC3:28;
     take d;
     thus 0 < d by A4;
     thus for x be Real st x in dom f & |. x-x0 .| < d holds |. f.x-f.x0 .|< e
     proof
      let x be Real;
      assume that
A5:    x in dom f and
A6:    |. x-x0 .| < d;
A7:   |. -(x-x0) .| = |. x-x0 .| by COMPLEX1:52;
      per cases;
      suppose x=x0;
       hence |. f.x-f.x0 .|< e by A3,COMPLEX1:44;
      end;
      suppose x<>x0; then
       x0-x <> 0;
       hence |. f.x-f.x0 .|< e by A4,A5,A6,A7,COMPLEX1:47;
      end;
     end;
    end;
    hence f is_continuous_in x0 by FCONT_1:3;
end;
