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,TOP-REAL m holds f is continuous iff
  for p being Point of R^1, r being positive Real
  ex s being positive Real st f.:].p-s,p+s.[ c= Ball(f.p,r)
  proof
    let f be Function of R^1, TOP-REAL m;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m by EUCLID:def 8;
    then reconsider f1 = f as Function of R^1,TopSpaceMetr Euclid m;
    hereby
      assume
A2:   f is continuous;
      let p be Point of R^1;
      let r be positive Real;
      reconsider p1 = p as Point of RealSpace;
      reconsider q1 = f.p as Point of Euclid m by EUCLID:67;
      f1 is continuous by A1,A2,YELLOW12:36;
      then consider s being positive Real such that
A3:   f1.:Ball(p1,s) c= Ball(q1,r) by Th17;
      take s;
      Ball(p1,s) = ].p-s,p+s.[ & Ball(q1,r) = Ball(f.p,r)
      by FRECHET:7,TOPREAL9:13;
      hence f.:].p-s,p+s.[ c= Ball(f.p,r) by A3;
    end;
    assume
A4: for p being Point of R^1, r being positive Real
    ex s being positive Real st f.:].p-s,p+s.[ c= Ball(f.p,r);
    for p being Point of RealSpace, q being Point of Euclid m,
    r being positive Real st q = f.p
    ex s being positive Real st f.:Ball(p,s) c= Ball(q,r)
    proof
      let p be Point of RealSpace, q be Point of Euclid m,
          r be positive Real such that
A5:   q = f.p;
      reconsider p1 = p as Point of R^1;
      consider s being positive Real such that
A6:   f.:].p-s,p+s.[ c= Ball(f.p1,r) by A4;
      take s;
      ].p-s,p+s.[ = Ball(p,s) & Ball(f.p1,r) = Ball(q,r)
      by A5,FRECHET:7,TOPREAL9:13;
      hence thesis by A6;
    end;
    then f1 is continuous by A1,Th17;
    hence thesis by A1,YELLOW12:36;
  end;
