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
  (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 RCOMP_1:16;
    f.:N = Im(f,x0) by A1,RELAT_1:112
      .= {f.x0} by A2,FUNCT_1:59;
    hence f.:N c= N1 by A3,ZFMISC_1:31;
  end;
  hence thesis by Th5;
end;
