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0 My textbook defines the space of all continuously differentiable functions C n ( X, E) as follows As such, f ∈ C n ( X, E) IFF the n th derivative of f is continuous on X I found that this definition coincides with one given by Wikipedia's page Roll's theorem is stated in my textbook as.
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Iand (1 p n)i!1 for all xed i Now, log lim n!1 (1 n=n)n = lim n!1 nlog(1 n=n) = lim n!1 n( n=n o(1=n)) (2) = lim n!1 n= So, lim. R ŃG N X e A, O \, , , , K f j O f G ɂ āB f U C H ܂ł ܂ B. A ~ n ځC X e X n ځC ` ^ n ځC n ڏC Ȃ n ډ com c S H ɂ C B A C f A ` ɂ 鎎 ̂ ` ܂ B.
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K W t W b N X E I g @ s n o ֎Ԃ 舵 Ă ܂ B ̃o C N T ܂ B C y ɂ ⍇ B ҂ Ă ܂ B. Jcnj converges, prove that the series f (x) ˘ X1 n˘1 cnI(x¡xn) (a •x •b) converges uniformly, and that f is continuous for every x 6˘xn Solution Let fk(x) ˘ Xk n˘1 cnI(x¡xn) By the Weierstrass Mtest (Theorem 710) with Mn ˘jcnj, {fk(x)} converges uniformly to f (x) Let E ˘ a,b\{xn n 2 N} Since each fk(x) is continuous on E, then by Theorem 712 we. N(x c)n Find a k Answer a k = f(k)(c) k!.
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