
theorem
for f be PartFunc of REAL,REAL, x0 be Real st
 dom f c= left_open_halfline x0 & f is_Lcontinuous_in x0
  holds f is_continuous_in x0
proof
    let f be PartFunc of REAL,REAL, x0 be Real;
    assume that
A1:  dom f c= left_open_halfline x0 and
A2:  f is_Lcontinuous_in x0;

    for s be Real_Sequence st rng s c= dom f & s is convergent & lim s = x0
     holds f/*s is convergent & f.x0 = lim(f/*s)
    proof
     let s be Real_Sequence;
     assume that
A3:   rng s c= dom f and
A4:   s is convergent and
A5:   lim s = x0;
     rng s c= left_open_halfline x0 by A1,A3; then
     rng s c= left_open_halfline x0 /\ dom f by A3,XBOOLE_1:19;
     hence f/*s is convergent & f.x0 = lim(f/*s) by A4,A5,A2,FDIFF_3:def 1;
    end;
    hence f is_continuous_in x0 by FCONT_1:def 1;
end;
