reserve
  n, m for Nat,
  T for non empty TopSpace,
  M, M1, M2 for non empty MetrSpace;

theorem
  for f being Function of T,TopSpaceMetr(M) holds f is continuous iff
  for p being Point of T, q being Point of M, r being positive Real
      st q = f.p
  ex W being open Subset of T st p in W & f.:W c= Ball(q,r)
  proof
    let f be Function of T,TopSpaceMetr(M);
    thus f is continuous implies
    for p being Point of T, q being Point of M, r being positive Real
    st q = f.p
    ex W being open Subset of T st p in W & f.:W c= Ball(q,r)
    proof
      assume
A1:   f is continuous;
      let p be Point of T;
      let q be Point of M;
      let r be positive Real;
      assume
A2:   f.p = q;
      reconsider V = Ball(q,r) as Subset of TopSpaceMetr(M);
A3:   q in Ball(q,r) by GOBOARD6:1;
      V is open by TOPMETR:14;
      then ex W being Subset of T st p in W & W is open & f.:W c= V
      by A1,A2,A3,JGRAPH_2:10;
      hence thesis;
    end;
    assume
A4: for p being Point of T, q being Point of M,
    r being positive Real st q = f.p
    ex W being open Subset of T st p in W & f.:W c= Ball(q,r);
    for p being Point of T, V being Subset of TopSpaceMetr(M)
    st f.p in V & V is open holds
    ex W being Subset of T st p in W & W is open & f.:W c= V
    proof
      let p be Point of T, V be Subset of TopSpaceMetr(M) such that
A5:   f.p in V;
      reconsider u = f.p as Point of M;
      assume V is open;
      then Int V = V by TOPS_1:23;
      then consider e being Real such that
A6:   e > 0 & Ball(u,e) c= V by A5,GOBOARD6:4;
      ex W being open Subset of T st p in W & f.:W c= Ball(u,e) by A4,A6;
      hence thesis by A6,XBOOLE_1:1;
    end;
    hence thesis by JGRAPH_2:10;
  end;
