reserve y for object, X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1 for Real;
reserve n,m,k for Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve h for non-zero 0-convergent Real_Sequence;
reserve c for constant Real_Sequence;
reserve R,R1,R2 for RestFunc;
reserve L,L1,L2 for LinearFunc;

theorem
  f is_differentiable_on X implies f|X is continuous
proof
  assume
A1: f is_differentiable_on X;
  let x be Real;
  assume x in dom(f|X);
  then x is Real & x in X;
  then f|X is_differentiable_in x by A1;
  hence thesis by Th24;
end;
