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

theorem
  for f being Function of R^1,T holds f is continuous iff
  for p being Point of R^1, V being open Subset of T st f.p in V
  ex s being positive Real st f.:].p-s,p+s.[ c= V
  proof
    let f be Function of R^1,T;
    hereby
      assume
A1:   f is continuous;
      let p be Point of R^1;
      let V be open Subset of T;
      assume f.p in V;
      then consider W being Subset of R^1 such that
A2:   p in W and
A3:   W is open and
A4:   f.:W c= V by A1,JGRAPH_2:10;
      consider s being Real such that
A5:   s > 0 and
A6:   ].p-s,p+s.[ c= W by A2,A3,FRECHET:8;
      reconsider s as positive Real by A5;
      take s;
      f.:].p-s,p+s.[ c= f.:W by A6,RELAT_1:123;
      hence f.:].p-s,p+s.[ c= V by A4;
    end;
    assume
A7: for p being Point of R^1, V being open Subset of T st f.p in V
    ex s being positive Real st f.:].p-s,p+s.[ c= V;
    for p being Point of R^1, V being Subset of T st f.p in V & V is open holds
    ex W being Subset of R^1 st p in W & W is open & f.:W c= V
    proof
      let p be Point of R^1, V be Subset of T such that
A8:   f.p in V and
A9:   V is open;
      consider s being positive Real such that
A10:   f.:].p-s,p+s.[ c= V by A7,A8,A9;
      take W = R^1(].p-s,p+s.[);
      thus p in W by TOPREAL6:15;
      thus W is open;
      thus thesis by A10;
    end;
    hence thesis by JGRAPH_2:10;
  end;
