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

    for s be Real_Sequence st rng s c= left_open_halfline x0 /\ 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= left_open_halfline x0 /\ dom f and
A4:   s is convergent and
A5:   lim s = x0;
     rng s c= dom f by A3,XBOOLE_1:18;
     hence f/*s is convergent & f.x0 = lim(f/*s) by A2,A4,A5,FCONT_1:def 1;
    end;
    hence f is_Lcontinuous_in x0 by A1,FDIFF_3:def 1;

    for s be Real_Sequence st rng s c= right_open_halfline x0 /\ 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
A6:   rng s c= right_open_halfline x0 /\ dom f and
A7:   s is convergent and
A8:   lim s = x0;
     rng s c= dom f by A6,XBOOLE_1:18;
     hence f/*s is convergent & f.x0 = lim(f/*s) by A2,A7,A8,FCONT_1:def 1;
    end;
    hence f is_Rcontinuous_in x0 by A1,FDIFF_3:def 2;
end;
