(1) Suppose f :(M, d) + (N,0) is not uniformly continuous. Show that there exist an...
Use the sequential criterion for the absence of uniform continuity to show that the function f(0, 1) - 1 given by f(x) = 1/-x) is not uniformly continuous. Theorem 4.4.5 (Sequential Criterion for Absence of Uniform Conti- nuity). A function /: A → R fails to be uniformly continuous on A if and only if there erists a particular to > 0 and two sequences (in) and (y) in A satisfying In - yn — but if(n)-f(n) Co.
Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if and only if fn(xn) → f(x) whenever xn → x. Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if...
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous on a compact set D. Then f is uniformly continuous on D. (a) Suppose that f is not uniformly continuous on D. Then there exists an for every n EN there exists xn and > 0 such that yn in D with la ,-ynl < 1/n and If(xn)-f(yn)12 E. (b) Apply 4.4.7, every bounded sequence has a convergent subsequence, to obtain a convergent subsequence...
Let f:D + R be a function. (a) Recall the definition that f is uniformly continuous on D. (You do not need to write this down. This only serves as a hint for next parts.) (b) Use (a) and the mean value theorem to prove f(x) = e-% + sin x is uniformly continuous on (0, +00). (c) Use the negation of (a) to prove f(x) = x2 is not uniformly continuous on (0,0).
3. A sequence is a map a N°R, typically written (an) = (ao, a1, a2, a3, a4,) As an example, the sequence (an) = 1/(n2 +1) begins (1, 1/2, 1/5, 1/10, 1/17,..) Here is a useful fact relating sequences and continuity: A function f(x) is continuous at x c if and only if for every sequence (an) that converges to c, written anc, then f(x,) f(c). Alternatively, if you and f(yn)L" with L' L", then f is not continuous at...
[4 Pts. Use the definition of continuity to show that the function f is continuous at <=0 10 g(x)= 3-4
Proposition 7.27. Suppose fn: G + C is continuous, for n > 1, (fn) converges uniformly to f :G+C, and y C G is a piecewise smooth path. Then lim n-00 $. fn = $. . 7.23. Let fn(x) = n2x e-nx. (a) Show that limn400 fn(x) = 0 for all x > 0. (Hint: Treat x = O as a special case; for x > 0 you can use L'Hôspital's rule (Theorem A.11) — but remember that n is...
Let f : [0,1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that 2 - y<== f() - f(y)< € for every 2, Y € [0,1]. The graph of f is the set Gj = {, f(c)): 1 € [0,1]}. : Show that G has measure zero.
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 (?),...
Let f: [0,1]→R be uniformly continuous, so that for every >0, there exists δ >0 such that |x−y|< δ=⇒|f(x)−f(y)|< for every x,y∈[0,1].The graph of f is the set G f={(x,f(x)) :x∈[0,1]}.Show that G f has measure zero Let f : [0, 1] → R be uniformly continuous, so that for every e > 0, there exists 8 >0 such that 2- y<83|f() - f(y)< € for every 1, 9 € [0,1]. The graph of f is the set Gj =...