Prove that f(x) is uniformly continuous on [0 inf) if lim f(x) = 0.
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 (?),...
Prove that f(x) = is uniformly continuous on (1,00) and not uniformly continuous on (0,1). (19 pts)
(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,
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.
(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
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...
(a) Suppose that lim x→c f(x) = L > 0. Prove that there exists a δ > 0 such that if 0 < |x − c| < δ, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on [a, b] and f(x) > 0 for all x ∈ [a, b], then there exists an " > 0 such that f(x) ≥ " for all x ∈ [a, b]. = (a) Suppose...
Prove that fx=5ln(x-7) is not 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 continuous on [0,∞).