
theorem
for f be PartFunc of REAL,REAL, x0 be Real st
 x0 in dom f & f is_right_convergent_in x0 & lim_right(f,x0) = f.x0
  holds 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_right_convergent_in x0 and
A3:  lim_right(f,x0) = f.x0;

    for a be Real_Sequence st rng a c= right_open_halfline x0 /\ dom f &
      a is convergent & lim a = x0 holds
       f/*a is convergent & f.x0 = lim(f/*a) by A2,A3,LIMFUNC2:def 8;
    hence f is_Rcontinuous_in x0 by A1,FDIFF_3:def 2;
end;
