reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th10:
  f is_continuous_in x0 & f.x0<>0 implies f^ is_continuous_in x0
proof
  assume that
A1: f is_continuous_in x0 and
A2: f.x0<>0;
  not f.x0 in {0} by A2,TARSKI:def 1;
  then
A3: not x0 in f"{0} by FUNCT_1:def 7;
  let s1;
  assume that
A4: rng s1 c= dom (f^) and
A5: s1 is convergent & lim s1=x0;
  dom f \ f"{0} c= dom f & rng s1 c= dom f \ f"{0} by A4,RFUNCT_1:def 2
,XBOOLE_1:36;
  then rng s1 c= dom f;
  then
A6: f/*s1 is convergent & f.x0 = lim (f/*s1) by A1,A5;
  then (f/*s1)" is convergent by A2,A4,RFUNCT_2:11,SEQ_2:21;
  hence (f^)/*s1 is convergent by A4,RFUNCT_2:12;
  x0 in dom f by A2,FUNCT_1:def 2;
  then x0 in dom f \ f"{0} by A3,XBOOLE_0:def 5;
  then x0 in dom (f^) by RFUNCT_1:def 2;
  hence (f^).x0 = (f.x0)" by RFUNCT_1:def 2
    .= lim ((f/*s1)") by A2,A4,A6,RFUNCT_2:11,SEQ_2:22
    .= lim ((f^)/*s1) by A4,RFUNCT_2:12;
end;
