reserve k, k1, n, n1, m for Nat;
reserve X, y for set;
reserve p for Real;
reserve r for Real;
reserve a, a1, a2, b, b1, b2, x, x0, z, z0 for Complex;
reserve s1, s3, seq, seq1 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f, f1, f2 for PartFunc of COMPLEX,COMPLEX;
reserve Nseq for increasing sequence of NAT;
reserve h for 0-convergent non-zero Complex_Sequence;
reserve c for constant Complex_Sequence;
reserve R, R1, R2 for C_RestFunc;
reserve L, L1, L2 for C_LinearFunc;
reserve Z for open Subset of COMPLEX;

theorem
  f is_differentiable_on X implies f is_continuous_on X
proof
  assume
A1: f is_differentiable_on X;
  hence
A2: X c= dom f;
  let x be Complex;
  assume x in X;
  then
A3: x in dom (f|X) by A2,RELAT_1:57;
  f|X is differentiable by A1;
  then f|X is_differentiable_in x by A3;
  hence thesis by Th34;
end;
