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
  f|X is continuous & (f|X)"{0} = {} implies f^|X is continuous
proof
  assume that
A1: f|X is continuous and
A2: (f|X)"{0} = {};
  f|X|X is continuous by A1;
  then (f|X)^|X is continuous by A2,Th22;
  then (f^)|X|X is continuous by RFUNCT_1:46;
  hence thesis;
end;
