3. Let the function f be a real valued bounded continuous function on R. Prove that...
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R. Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists M> 0 such that n(x) M for all r E [0,1] and all n N. (b) Does the result in part (a) hold if uniform convergence is replaced by pointwise convergence? Prove or give a counterexample (2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists...
Number 6 please S. Let ) be a sequence of continuous real-valued functions that converges uniformly to a function fon a set ECR. Prove that lim S.(z) =S(x) for every sequence (x.) C Esuch that ,E E 6. Let ECRand let D be a dense subset of E. If .) is a sequence of continuous real-valued functions on E. and if () converges unifomly on D. prove that (.) converges uniformly on E. (Recall that D is dense in E...
(5) Let f: [0, 1 R. We say that f is Hölder continuous of order a e (0,1) if \f(x) -- f(y)| . , y sup [0, 1] with 2 # 1£l\c° sup is finite. We define Co ((0, 1]) f: [0, 1] -R: f is Hölder continuous of order a}. = (a) For f,gE C ([0, 1]) define da(f,g) = ||f-9||c«. Prove that da is a well-defined metric Ca((0, 1) (b) Prove that (C ([0, 1]), da) is complete...
Proof Theorem 65.6 (a generalization of Dini's theorem) Let {fn be a sequence of real-valued continuous functions on a compact subset S of R such that (1) for each x € S, the sequenсe {fn(x)}o is bounded and топotone, and (ii) the function x lim,0 fn(x) is continuous on S Then f Remark that the result is not always true without the monotonicity of item (i) Šn=0 lim fn uniformly on S Theorem 65.6 (a generalization of Dini's theorem) Let...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
Can someone show me how to do question 2a and all 3 and 4? I tried ratio test for 2a, but if x = 0, rhe proof doesn't work. Thanks a lot. 2. Prove the following. (a) The series o converges for all 3 € R. (b) For n e N and k € {2,..., n}, the binomial coefficient (7) satisfies *)-(-5) (-)-(---) (c) For x > 0, the sequence (1 + 5)" is monotone increasing and bounded above by...
0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer the following questions value A as Find the following limit lim aoo a2 f (x)dx 0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer the following questions value A as Find the following limit lim aoo a2 f (x)dx
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x E A, we have lÍs(x)-f(x)|く for all n > N. Then f is uniformly continuous. Note: We could say that the "sequence of functions" f "converges to the function" f. These are not defined terms for...
(Exercise 5.8 of the text) If f is a real-valued function on (a, b) and c E (a, b), we say that f is strictly increasing at c if there exists δ 〉 0 such that if c-5 〈 x 〈 c, then f(x) 〈 f(c) and if c 〈 x 〈 c+6, then f(c) 〈 f(x). We say that f is strictly increasing on (a, b) if whenever x, y E (a, b) with x 〈 y, we have...