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

theorem
  for f being Function of TopSpaceMetr(M),T holds f is continuous iff
  for p being Point of M, V being open Subset of T st f.p in V
  ex s being positive Real st f.:Ball(p,s) c= V
  proof
    let f be Function of TopSpaceMetr(M),T;
    hereby
      assume
A1:   f is continuous;
      let p be Point of M;
      let V be open Subset of T;
      assume f.p in V;
      then consider W being Subset of TopSpaceMetr(M) such that
A2:   p in W and
A3:   W is open and
A4:   f.:W c= V by A1,JGRAPH_2:10;
      Int W = W by A3,TOPS_1:23;
      then consider s being Real such that
A5:   s > 0 and
A6:   Ball(p,s) c= W by A2,GOBOARD6:4;
      reconsider s as positive Real by A5;
      take s;
      f.:Ball(p,s) c= f.:W by A6,RELAT_1:123;
      hence f.:Ball(p,s) c= V by A4;
    end;
    assume
A7: for p being Point of M, V being open Subset of T st f.p in V
    ex s being positive Real st f.:Ball(p,s) c= V;
    for p being Point of TopSpaceMetr(M), V being Subset of T
    st f.p in V & V is open holds
    ex W being Subset of TopSpaceMetr(M) st p in W & W is open & f.:W c= V
    proof
      let p be Point of TopSpaceMetr(M), V be Subset of T such that
A8:   f.p in V and
A9:   V is open;
      reconsider u = p as Point of M;
      consider s being positive Real such that
A10:   f.:Ball(u,s) c= V by A7,A8,A9;
      reconsider W = Ball(u,s) as Subset of TopSpaceMetr(M);
      take W;
      thus p in W by GOBOARD6:1;
      thus W is open by TOPMETR:14;
      thus thesis by A10;
    end;
    hence thesis by JGRAPH_2:10;
  end;
