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,R^1 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= ].f.p-r,f.p+r.[
  proof
    let f be Function of R^1,R^1;
    hereby
      assume
A1:   f is continuous;
      let p be Point of R^1;
      let r be positive Real;
      reconsider p1 = p, q1 = f.p as Point of RealSpace;
      consider s being positive Real such that
A2:   f.:Ball(p1,s) c= Ball(q1,r) by A1,Th17;
      take s;
      Ball(p1,s) = ].p1-s,p1+s.[ & Ball(q1,r) = ].q1-r,q1+r.[
      by FRECHET:7;
      hence f.:].p-s,p+s.[ c= ].f.p-r,f.p+r.[ by A2;
    end;
    assume
A3: for p being Point of R^1, r being positive Real
    ex s being positive Real st f.:].p-s,p+s.[ c= ].f.p-r,f.p+r.[;
    for p, q being Point of RealSpace, 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, q be Point of RealSpace,
          r be positive Real such that
A4:   q = f.p;
      consider s being positive Real such that
A5:   f.:].p-s,p+s.[ c= ].f.p-r,f.p+r.[ by A3;
      take s;
      Ball(p,s) = ].p-s,p+s.[ & Ball(q,r) = ].q-r,q+r.[ by FRECHET:7;
      hence thesis by A5,A4;
    end;
    hence f is continuous by Th17;
  end;
