a) Prove that for all x, y≥0 we have |√x−√y|≤√|x-y|.
(b) Prove that f(x)=√x is uniformly continuous on [0,∞).
a) Prove that for all x, y≥0 we have |√x−√y|≤√|x-y|. (b) Prove that f(x)=√x is uniformly...
Prove that f(x) is uniformly continuous on [0 inf) if lim f(x) = 0.
8. *** Prove that f(x) = ? is not uniformly continuous on (0,0). Remark: Recall that we proved in class that f(0) = .za is not uniformly continuous on (0,0). I remind you of this result in case it helps you think about how to approach this exercise.
2. a. Prove that f(x) = V22 - 13 is uniformly continuous on the interval (7,0). b. Prove or disprove: f(x) = V x2 - 13 is not uniformly continuous on the interval ( 13, 7). c. Prove or disprove: If a > 13, f(x) = 32-13 is uniformly continuous on the interval (a,
(b) 4 Let F: X Y be a linear map between two normed spaces. Prove that F is continuous at Ojf and only if F is uniformly continuous on X.
Prove that f(x) = is uniformly continuous on (1,00) and not uniformly continuous on (0,1). (19 pts)
3. Let f : (a,b) +R be a function such that for all x, y € (a, b) and all t € (0.1) we have (tx + (1 - t)y)<tf(x) + (1 - t)f(y). Prove that f is continuous on (a,b).
(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...
(b) Let a >0. Does (f.) converge uniformly on [-a, al? (c) Does (f) converge uniformly on R? Q4 You are given the series n2 +r2 (a) Prove that the series converges uniformly on [-a, al for each a > 0. (b) Prove that the sum F(r) is well defined and continuous on R. (c) Prove that the series does not converge uniformly on R. Q5 You are given the series I n2r2 (b) Let a >0. Does (f.) converge...
Definition: A function f : A → R is said to be uniformly continuous on A if for every e > O there is a δ > 0 such that *for all* z, y € A we have Iz-vl < δ nnplies If(r)-f(y)| < e. In other words a function is uniformly continuous if it is continuous at every point of its domain (e.g. every y A), with the delta response to any epsilon challenge not depending on which point...
Urgent help needed in Math Problems ! Thanx 3. Prove that f(x)=1/(1-) is not uniformly continuous for 12 <1. 4. Show that the function f(x) = 1/22 is not uniformly continuous for 0 < Rez <1/2 but is uniformly continuous for 1/2 < Rez < 1. 6. Discuss continuity of (Rez)? (Im ) if : +0 if 20 f(2)= |z| 2 my 0 if = 0 at the all points of C. 7. Find the following limits: (a) lim (?),...