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

theorem
  x0 in dom f & (ex N be Neighbourhood of x0 st dom f /\ N = {x0})
  implies f is_continuous_in x0
proof
  assume
A1: x0 in dom f;
  given N be Neighbourhood of x0 such that
A2: dom f /\ N = {x0};
  now
    let N1 be Neighbourhood of f/.x0;
    take N;
A3: f/.x0 in N1 by Th4;
    f.:N = Im(f,x0) by A2,RELAT_1:112
      .= {f.x0} by A1,FUNCT_1:59
      .= {f/.x0} by A1,PARTFUN1:def 6;
    hence f.:N c= N1 by A3,ZFMISC_1:31;
  end;
  hence thesis by A1,Th10;
end;
