[加上均匀收敛(uniform.doc

Chaper7 Infinite series of functions Section 7.1 Uniform Convergence of Sequence Definition .Let and let . (1).. (2). Definition 7.1. Let ,. We say that converges to f if Notation: Remark. . Remark 7.3. Noting that discontinuous. Remark 7.4. Let For examples, Noting that Remark7.6: (3) Proof. Then . Question differentiable Is f differentiable ? Remark7.3 is differentiable differentiable at 1. Remark7.5: differentiable But 結論 differentiable Is f differentiable ? No differentiable and is f differentiable ? No integrable and [加上均勻收斂(uniform convergence)才會對] Definition 7.7: Let , . We say that converges uniformly to f if Notation : Example 7.8: Proof. Exsmple Proof . Theorem7.9: . Proof. . . , We have . Remark: and Theorem7.10. Then Proof. (1) . . For this P, (2) Lemma7.11:,if and only if . Proof. Theorem 7.12: Proof. differentiable differentiable Hence, by (*) and Exercise 3(c) , . (2)differentiable . On the other hand, (3) Repeating the produce of (1) and (2) We know that differentiable and .