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

theorem
for f be PartFunc of REAL,REAL, I be non empty Interval st
 f is_differentiable_on_interval I holds f|I is continuous
proof
    let f be PartFunc of REAL,REAL, I be non empty Interval;
    assume
A1:  f is_differentiable_on_interval I;

    for x be Real st x in dom(f|I) holds f|I is_continuous_in x
    proof
     let x be Real;
     assume
A2:   x in dom(f|I);
     per cases;
     suppose
A3:   x = inf I; then
      f|I is_right_differentiable_in x by A1,A2,Th11; then
A4:   (f|I)|(right_closed_halfline x) is_continuous_in x
        by Th35,FDIFF_3:7;
      now let y be Real;
       assume y in I; then
       x <= y by A3,XXREAL_2:3; then
       y in [.x,+infty.[ by XXREAL_1:236;
       hence y in right_closed_halfline x by LIMFUNC1:def 2;
      end; then
      I c= right_closed_halfline x;
      hence f|I is_continuous_in x by A4,RELAT_1:73;
     end;
     suppose
A5:   x = sup I; then
      f|I is_left_differentiable_in x by A1,A2,Th11; then
A6:   (f|I)|(left_closed_halfline x) is_continuous_in x
        by Th36,FDIFF_3:5;
      now let y be Real;
       assume y in I; then
       y <= x by A5,XXREAL_2:4; then
       y in ].-infty,x.] by XXREAL_1:234;
       hence y in left_closed_halfline x by LIMFUNC1:def 1;
      end; then
      I c= left_closed_halfline x;
      hence f|I is_continuous_in x by A6,RELAT_1:73;
     end;
     suppose x <> inf I & x <> sup I; then
      x in ].inf I,sup I.[ by A2,Th3; then
      f is_differentiable_in x by A1,A2,Th11;
      hence f|I is_continuous_in x by A2,FDIFF_1:24,FCONT_1:1;
     end;
    end;
    hence f|I is continuous by FCONT_1:def 2;
end;
