reserve f,g for PartFunc of REAL,REAL,
  r,r1,r2,g1,g2,g3,g4,g5,g6,x,x0,t,c for Real,
  a,b,s for Real_Sequence,
  n,k for Element of NAT;

theorem
  f is_continuous_in x0 & (for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1
& g1<x0 & g1 in dom f & g2<r2 & x0<g2 & g2 in dom f) implies f is_convergent_in
  x0
proof
  assume that
A1: f is_continuous_in x0 and
A2: for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom f &
  g2<r2 & x0<g2 & g2 in dom f;
  now
    let s such that
A3: s is convergent & lim s=x0 and
A4: rng s c= dom f \ {x0};
    dom f \ {x0} c= dom f by XBOOLE_1:36;
    then rng s c= dom f by A4;
    hence f/*s is convergent & lim (f/*s) = f.x0 by A1,A3,FCONT_1:def 1;
  end;
  hence thesis by A2,LIMFUNC3:def 1;
end;
