reserve n,m,k for Nat;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,g,x0,x1,x2 for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of REAL,the carrier of S;
reserve s1,s2 for Real_Sequence;
reserve Y for Subset of REAL;

theorem Th10:
for S,f,x0 holds
 f is_continuous_in x0
iff x0 in dom f
  & for N1 being Neighbourhood of f/.x0
      ex N being Neighbourhood of x0 st f.:N c= N1
proof
   let S,f,x0;
   thus f is_continuous_in x0 implies
     x0 in dom f & for N1 being Neighbourhood of f/.x0
      ex N being Neighbourhood of x0 st f.:N c= N1
   proof
    assume A1: f is_continuous_in x0;
    hence x0 in dom f;
    let N1 be Neighbourhood of f/.x0;
    consider N being Neighbourhood of x0 such that
A2:  for x1 st x1 in dom f & x1 in N holds f/.x1 in N1 by A1,Th9;
    take N;
    now let r be object;
     assume r in f.:N; then
     consider x be Element of REAL such that
A3:   x in dom f & x in N & r = f.x by PARTFUN2:59;
     r=f/.x by A3,PARTFUN1:def 6;
     hence r in N1 by A2,A3;
    end;
    hence thesis by TARSKI:def 3;
   end;
   assume
A4: x0 in dom f
  & for N1 being Neighbourhood of f/.x0
      ex N being Neighbourhood of x0 st f.:N c= N1;
   now let N1 be Neighbourhood of f/.x0;
    consider N being Neighbourhood of x0 such that
A5:  f.:N c= N1 by A4;
    take N;
    let x1;
    assume A6: x1 in dom f & x1 in N; then
    f.x1 in f.:N by FUNCT_1:def 6; then
    f/.x1 in f.:N by A6,PARTFUN1:def 6;
    hence f/.x1 in N1 by A5;
   end;
   hence thesis by A4,Th9;
end;
