Urgent help needed in Math Problems ! Thanx
As not mentioned which question has to be answered the answer of the first question is provided.
Urgent help needed in Math Problems ! Thanx 3. Prove that f(x)=1/(1-) is not uniformly continuous...
Prove that f(x) is uniformly continuous on [0 inf) if lim f(x) = 0.
9. Prove that the function f(x) = ax+b is uniformly continuous on R by directly applying the e, 8 definition of uniform continuity.
Use the definition of uniform continuity to prove that f(x)is uniformly continuous on , 00
(10) Prove that if (fn) is a sequence of uniformly continuous functions on the interval (a, b) such (a, b), then f is also uniformly continuous on (a, b) that f funiformly on en dr 0. (11) Show that lim n-+o0 e (10) A G.. 11d
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.
(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 f is uniformly continuous on fn(x)=f(x+1/n) converges uniformly to f on 4.3.1. Using Exercise 3.3.22, show that n! k -w (k-1)(n- k)! (1 -2)"-k dz w=k where 0< p<1, and k and n are positive integers such that k < n. 4.3.1. Using Exercise 3.3.22, show that n! k -w (k-1)(n- k)! (1 -2)"-k dz w=k where 0
(10 marks) Prove that fx=6ln(x-11) is not uniformly continuous on (0,∞) Х Enable Editing X i PROTECTED VIEW Be careful—files from the Internet can contain viruses. Unless you need to edit, it's safer to stay in Protected View. LAAM Yuuuus = (x2-x-2 1. (10 marks) Let f(x) (x2-4) if x # +2 с if x = 2 Find c that would make f continuous at 1. For such c, prove that f is continuous at 1 using an ε -...
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,
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...