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
(ex N be Neighbourhood of x0 st dom f /\ N = {x0})
  implies f is_continuous_in x0
proof
   given N be Neighbourhood of x0 such that
A1: dom f /\ N = {x0};
   x0 in dom f /\ N by A1,TARSKI:def 1; then
A2:x0 in dom f by XBOOLE_0:def 4;
   now let N1 be Neighbourhood of f/.x0;
    take N;
A3: f/.x0 in N1 by NFCONT_1:4;
    f.:N = Im(f,x0) by A1,RELAT_1:112
        .= {f.x0} by A2,FUNCT_1:59
        .= {f/.x0} by A2,PARTFUN1:def 6;
    hence f.:N c= N1 by A3,ZFMISC_1:31;
   end;
   hence thesis by Th10,A2;
end;
