reserve X for RealNormSpace;

theorem
  for X,Y be RealNormSpace, f be PartFunc of X,Y, ft be Function of
TopSpaceNorm X,TopSpaceNorm Y st f = ft holds f is_continuous_on the carrier of
  X iff ft is continuous
proof
  let X,Y be RealNormSpace, f be PartFunc of X,Y, ft be Function of
  TopSpaceNorm X,TopSpaceNorm Y;
  assume
A1: f=ft;
A2: f|(the carrier of X)=f by RELSET_1:19;
  hereby
    assume
A3: f is_continuous_on the carrier of X;
    now
      let xt be Point of TopSpaceNorm X;
      reconsider x=xt as Point of X;
      f|(the carrier of X) is_continuous_in x by A3,NFCONT_1:def 7;
      hence ft is_continuous_at xt by A1,A2,Th18;
    end;
    hence ft is continuous by TMAP_1:44;
  end;
  assume
A4: ft is continuous;
A5: now
    let x be Point of X;
    assume x in the carrier of X;
    reconsider xt=x as Point of TopSpaceNorm X;
    ft is_continuous_at xt by A4,TMAP_1:44;
    hence f|(the carrier of X) is_continuous_in x by A1,A2,Th18;
  end;
  dom f =the carrier of X by A1,FUNCT_2:def 1;
  hence thesis by A5,NFCONT_1:def 7;
end;
