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

theorem
  for f being Function of TOP-REAL m,T holds f is continuous iff
  for p being Point of TOP-REAL 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 TOP-REAL m,T;
A1: m in NAT by ORDINAL1:def 12;
    hereby
      assume
A2:   f is continuous;
      let p be Point of TOP-REAL m;
      let V be open Subset of T;
      assume f.p in V;
      then consider W being Subset of TOP-REAL m such that
A3:   p in W and
A4:   W is open and
A5:   f.:W c= V by A2,JGRAPH_2:10;
      reconsider u = p as Point of Euclid m by EUCLID:67;
      Int W = W by A4,TOPS_1:23;
      then consider s being Real such that
A6:   s > 0 and
A7:   Ball(u,s) c= W by A3,GOBOARD6:5;
      reconsider s as positive Real by A6;
      take s;
      Ball(u,s) = Ball(p,s) by TOPREAL9:13;
      then f.:Ball(p,s) c= f.:W by A7,RELAT_1:123;
      hence f.:Ball(p,s) c= V by A5;
    end;
    assume
A8: for p being Point of TOP-REAL 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 TOP-REAL m, V being Subset of T
    st f.p in V & V is open holds
    ex W being Subset of TOP-REAL m st p in W & W is open & f.:W c= V
    proof
      let p be Point of TOP-REAL m, V be Subset of T such that
A9:   f.p in V and
A10:   V is open;
      consider s being positive Real such that
A11:   f.:Ball(p,s) c= V by A8,A9,A10;
      take W = Ball(p,s);
      thus p in W by A1,TOPGEN_5:13;
      thus W is open;
      thus thesis by A11;
    end;
    hence thesis by JGRAPH_2:10;
  end;
