R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous...
(1) Suppose f :(M, d) + (N,0) is not uniformly continuous. Show that there exist an a > 0 and sequences (Xn) and (yn) in M such that d(Ion, yn) < and o(f(xn), f(n)) > € VnE N. (Hint: Negation of the definition of uniform continuity.)
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...
Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R. Suppose that, for each the sequence (fe(x))ke N 1s a monotonic sequence which converges to (x). Show that r є X, k)kEN Converges to j uniformly. Let (X, d) be a compact metric space, and con- sider continuous functions fk : X → R, for k N, and f : X → R....
Problem 1. Suppose that f:(a,b) + R is a continuous function and there exists a point p e (a, b) such that f' exists and is bounded on (a,b) {p}. Prove that f is uniformly continuous on (a,b).
Real Analysis II Please do it without using Heine-Borel's theorem and do it only if you're sure Problem: Let E be a closed bounded subset of En and r be any function mapping E to (0,∞). Then there exists finitely many points yi ∈ E, i = 1,...,N such that Here Br(yi)(yi) is the open ball (neighborhood) of radius r(yi) centered at yi. Also, following definitions & theorems should help that E CUBy Definition. A subset S of a topological...
Please do 10 & 11 Use Intermediate Value Theorem lial p(x) = x4 +7x = 9 has two real root. 8. df Open with Google Docs Then use your calculator to find the ro 9 Let f(z)2with € [0, 00). Find a positive mumber e and two sequences {xn} and {yn} such that lim-(nn) = 0 but |f(xn)- f(Yn)| 2 e. Then conclude that f(x) = x2 is not uniformly continuous on [0, ao) [0, oo). Show that f is...
3. Suppose f : [0,) + R is a continuous function and that L limf(x) exists is a real number). Prove that f is uniformly continuous on (0,.). Suggestion: Let e > 0. Write out what the condition L = lim,+ f(t) means for this e: there erists M > 0 such that... Also write out what you are trying to prove about this e in this problem. Note that f is uniformly continuous on (0.M +1] because this is...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
Determine whether the statement is TRUE or FALSE. You are NOT required to justify your answers. (a) Suppose both f and g are continuous on (a, b) with f > 9. If Sf()dx = Sº g(x)dx, then f(x) = g(x) for all 3 € [a, b]. (b) If f is an infinitely differentiable function on R with f(n)(0) = 0 for all n = 0,1,2,..., then f(x) = 0 for all I ER. (c) f is improperly integrable on (a,...
2. Let DCR, and suppose the functions f:D + R and g: D R are continuous at to ED. Use the - definition of continuity to prove that f+g and fg are both continuous at ro e D: that is, prove that for every e > 0 there exists 8 >0 such that (+9)(2)-(+9)(20) <and (9)(x) - (49)(20) << whenever re D and r-rol < 8. (Hint. Use inequalities similar to those we used to prove the cor- responding results...