
theorem
for x0 be Real, f be PartFunc of REAL,REAL st
 x0 in dom f & f is_left_convergent_in x0 & lim_left(f,x0) = f.x0
  holds f is_Lcontinuous_in x0
proof
    let x0 be Real, f be PartFunc of REAL,REAL;
    assume that
A1:  x0 in dom f and
A2:  f is_left_convergent_in x0 and
A3: lim_left(f,x0) = f.x0;

    for seq be Real_Sequence st rng seq c= left_open_halfline(x0) /\ dom f &
     seq is convergent & lim seq = x0
     holds f/*seq is convergent & f.x0 = lim(f/*seq) by A2,A3,L_HOSPIT:3;
    hence f is_Lcontinuous_in x0 by A1,FDIFF_3:def 1;
end;
