reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve X for set;
reserve x,x0,g,r,s,p 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 seq for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,the carrier of F;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve R,R1,R2 for RestFunc of F;
reserve L,L1,L2 for LinearFunc of F;

theorem
  f is_differentiable_on X implies (f|X) is continuous
  proof
    assume
    A1: f is_differentiable_on X;
     for x being Real st x in dom(f|X) holds f|X is_continuous_in x
          by A1,Th22;
    hence thesis by NFCONT_3:def 2;
  end;
