推 hjmeric :第四行 第二個[a,b]應該要改成 (a,b)會較好吧 03/24 11:19
※ 編輯: zhanguihan 來自: 140.112.104.65 (03/24 11:26)
→ ej001 :as follow 03/24 15:07
※ 引述《iddee ()》之銘言：
: 對的話證明之，錯則給反例。
: 設 f_n(x): [a,b] -> R 在 [a,b] 上連續且在 (a,b) 上可微
: 若 f_n 均勻收斂至 f
: 則 f 在 (a,b) 上可微
This statement is not ture. The counterexample can be constructed as
following:
we know there exists an function defined on [a,b] which is continuous
on [a,b] but is not differentiable on (a,b). On the other hand, any
continuous function on a closed and bounded interval can be uniformly
approximated by the polynomials ( Weierstrass approximation theorem).
Hence, the counterexample exists.
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