 reserve h,h1 for 0-convergent non-zero Real_Sequence,
         c,c1 for constant Real_Sequence;

theorem
  for X be open Subset of REAL, f,F be PartFunc of REAL,REAL st
   X c= dom f & f|X is continuous holds
    for x be Real st x in X holds f is_continuous_in x
proof
    let X be open Subset of REAL, f,F be PartFunc of REAL,REAL;
    assume that
A1:  X c= dom f and
A2:  f|X is continuous;

A3: dom(f|X) = X by A1,RELAT_1:62;

    hereby let x be Real;
     assume
A4:   x in X;

     for r be Real st 0 < r
      ex s be Real st 0 < s & for x1 be Real st x1 in dom f & |.x1-x.| < s
       holds |.f.x1 - f.x.| < r
     proof
      let r be Real;
      consider ss1 be Real such that
A5:   0 < ss1 and
A6:   ].x-ss1,x+ss1.[ c= X by A4,RCOMP_1:19;
      assume 0 < r; then
      consider s be Real such that
A7:   0 < s and
A8:   for x1 be Real st x1 in dom(f|X) & |.x1-x.|<s holds
        |.(f|X).x1 - (f|X).x.| < r by A2,A4,A3,FCONT_1:def 2,3;
      set s1=min(ss1,s);
      take s1;
      now let x1 be Real;
       assume that
        x1 in dom f and
A9:    |.x1-x.|<s1;
       s1 <= s by XXREAL_0:17; then
A10:   |.x1-x.| < s by A9,XXREAL_0:2;
       s1 <= ss1 by XXREAL_0:17;
       then |.x1-x.| < ss1 by A9,XXREAL_0:2;
       then
A11:   x1 in ].x-ss1,x+ss1.[ by RCOMP_1:1;
       then |.f.x1-f.x.| = |.(f|X).x1 - f.x.| by A6,A3,FUNCT_1:47;
       then |.f.x1-f.x.| = |.(f|X).x1 - (f|X).x.| by A4,A3,FUNCT_1:47;
       hence |.f.x1-f.x.|<r by A8,A6,A10,A11,A3;
      end;
      hence thesis by A7,A5,XXREAL_0:15;
     end;
     hence f is_continuous_in x by FCONT_1:3;
    end;
end;
