reserve r for Real,
  X for set,
  f, g, h for real-valued Function;
reserve T for non empty TopSpace,
  A for closed Subset of T;

theorem Th19:
  for f being Function of T,R^1, x being Point of T holds f
is_continuous_at x iff for e being Real st e>0 ex H being Subset of T st
  H is open & x in H & for y being Point of T st y in H holds |.f.y-f.x.|<e
proof
  let f be Function of T,R^1, x be Point of T;
  thus f is_continuous_at x implies for e being Real st e>0 ex H being
Subset of T st H is open & x in H & for y being Point of T st y in H
   holds |.f.y-f.x.|<e
  proof
    reconsider fx=f.x as Point of RealSpace by TOPMETR:12,def 6;
    assume
A1: f is_continuous_at x;
    let e be Real such that
A2: e>0;
    reconsider G=Ball(fx,e) as Subset of R^1 by TOPMETR:12,def 6;
    G is open & fx in G by A2,GOBOARD6:1,TOPMETR:14,def 6;
    then consider H being Subset of T such that
A3: H is open & x in H and
A4: f.:H c= G by A1,TMAP_1:43;
    take H;
    thus H is open & x in H by A3;
A5: dom f=the carrier of T by FUNCT_2:def 1;
    let y be Point of T;
    assume y in H;
    then
A6: f.y in f.:H by A5,FUNCT_1:def 6;
    then f.y in G by A4;
    then reconsider fy=f.y as Point of RealSpace;
    dist(fy,fx) < e by A4,A6,METRIC_1:11;
    hence thesis by TOPMETR:11;
  end;
  assume
A7: for e being Real st e>0 ex H being Subset of T st H is open
  & x in H & for y being Point of T st y in H holds |.f.y-f.x.|<e;
  now
    reconsider fx=f.x as Point of RealSpace by TOPMETR:12,def 6;
    let G be Subset of R^1;
    assume G is open & f.x in G;
    then consider r being Real such that
A8: r>0 and
A9: Ball(fx,r) c= G by TOPMETR:15,def 6;
    consider H being Subset of T such that
A10: H is open & x in H and
A11: for y being Point of T st y in H holds |.f.y-f.x.|<r by A7,A8;
    take H;
    thus H is open & x in H by A10;
    thus f.:H c= G
    proof
      let a be object;
      assume a in f.:H;
      then consider y being object such that
A12:  y in dom f and
A13:  y in H and
A14:  a=f.y by FUNCT_1:def 6;
      reconsider y as Point of T by A12;
      reconsider fy=f.y as Point of RealSpace by TOPMETR:12,def 6;
      |.f.y-f.x.|<r by A11,A13;
      then |.-(f.y-f.x).|< r by COMPLEX1:52;
      then |.f.x-f.y.|<r;
      then dist(fx,fy)<r by TOPMETR:11;
      then fy in Ball(fx,r) by METRIC_1:11;
      hence thesis by A9,A14;
    end;
  end;
  hence thesis by TMAP_1:43;
end;
