reserve n,m for Nat;
reserve x,X,X1 for set;
reserve s,g,r,p for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1,s2 for sequence of S;
reserve x0,x1,x2 for Point of S;
reserve Y for Subset of S;

theorem Th19:
  f is_continuous_on X iff X c= dom f & for x0,r st x0 in X & 0<r
  ex s st 0<s & for x1 st x1 in X & ||. x1- x0 .|| < s holds ||. f/.x1 - f/.x0
  .|| < r
proof
  thus f is_continuous_on X implies X c= dom f & for x0,r st x0 in X & 0<r ex
s st 0<s & for x1 st x1 in X & ||. x1- x0 .|| < s holds ||. f/.x1 - f/.x0 .|| <
  r
  proof
    assume
A1: f is_continuous_on X;
    hence X c= dom f;
A2: X c= dom f by A1;
    let x0,r;
    assume that
A3: x0 in X and
A4: 0<r;
    f|X is_continuous_in x0 by A1,A3;
    then consider s such that
A5: 0<s and
A6: for x1 st x1 in dom(f|X) & ||. x1- x0 .|| <s holds ||. (f|X)/.x1-(
    f| X)/.x0 .||<r by A4,Th7;
    take s;
    thus 0<s by A5;
    let x1;
    assume that
A7: x1 in X and
A8: ||. x1- x0 .|| <s;
A9: dom (f|X) = dom f /\ X by PARTFUN2:15
      .= X by A2,XBOOLE_1:28;
    then ||. f/.x1 - f/.x0 .|| = ||. (f|X)/.x1 - f/.x0 .|| by A7,PARTFUN2:15
      .= ||. (f|X)/.x1 - (f|X)/.x0 .|| by A3,A9,PARTFUN2:15;
    hence thesis by A6,A9,A7,A8;
  end;
  assume that
A10: X c= dom f and
A11: for x0,r st x0 in X & 0<r ex s st 0<s & for x1 st x1 in X & ||. x1-
  x0 .|| < s holds ||. f/.x1 - f/.x0 .|| < r;
A12: dom (f|X) = dom f /\ X by PARTFUN2:15
    .= X by A10,XBOOLE_1:28;
  now
    let x0 such that
A13: x0 in X;
    for r st 0<r ex s st 0<s & for x1 st x1 in dom(f|X) & ||. x1- x0 .||
    <s holds ||. (f|X)/.x1-(f|X)/.x0 .||<r
    proof
      let r;
      assume 0<r;
      then consider s such that
A14:  0<s and
A15:  for x1 st x1 in X & ||. x1- x0 .|| < s holds ||. f/.x1 - f/.x0
      .|| < r by A11,A13;
      take s;
      thus 0<s by A14;
      let x1 such that
A16:  x1 in dom(f|X) and
A17:  ||. x1- x0 .|| <s;
      ||. (f|X)/.x1-(f|X)/.x0.|| = ||. (f|X)/.x1 - f/.x0.|| by A12,A13,
PARTFUN2:15
        .= ||. f/.x1 - f/.x0 .|| by A16,PARTFUN2:15;
      hence thesis by A12,A15,A16,A17;
    end;
    hence f|X is_continuous_in x0 by A12,A13,Th7;
  end;
  hence thesis by A10;
end;
