2. (a) Let M > 0 and RR satisfy the condition f(x)/(v) ≤ May for all x, y E R. Show that is uniformly continuous on R. [5 marks]

Q: 2. Consider the function / with domain (-2, o0): 3|2 – 1| 2 + 27 - 4 if – 2 <z<1 f(z) = if r = 1…

A:

Q: Prove: Let the function f:X →Y and g:Y →Z be continuous. Then the composition function g•f:X→Z is…

A:

Q: 3. Let f, g to be two continuous functions on [a, b] and f(a) ≥ g(a), f(b) ≤ g(b). Prove that Exo €…

A:

Q: b) Give an example to show that the product fg need not be uniformly continuous.

A: Given: f:A→B, g:B→C are uniformly continuous.To show: b) The product fg need not be uniformly…

Q: 10. Prove the following change of variables formula: Let p be a real-valued differentiable function…

A: Introduction: The department of calculus wherein we look at integrals and their houses is called…

Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

A:

Q: 3. Let A be the set of all functions ƒ : 0, 1] → R that are continuous in the interval [0, 1].…

A:

Q: 11. with f(-1)=6, f(1) = 8 and f(6) = 3. Intermediate Value Theorem? Let f be a function that is…

A: Given:f be a function that is differentiable on the closed interval -1, 6 such…

Q: 3c. Let 0 < a < 1 and ƒ : (a, 2) → R be given by ƒ (x) = ½. Show that ƒ is uniformly continuous.

A: Given: Let  and  be given by We have to show that  is uniformly continuous.

Q: (a) Assume f: S → R is uniformly continuous and S is bounded. Prove that the range of f is bounded.…

A:

Q: Let I be an interval and B a subset of R. Let f:I → B be a function continuous on I. AIM: To test…

A:

Q: 15) Prove that if ffx) is continuous at x=0 then so is V f(x) provided that fix) zo for all x in the…

A: Step 1: Step 2: Step 3: Step 4:

Q: Let K> 0 and let f: RR satisfy the condition f(x)-f(y)| ≤ Kx-y for all x, y € R. Show that f is…

A:

Q: 3. Let z € R" be denoted by x = (₁, 2,...,n) and || x ||:= √√x² + x² + +x². Let f: RR be defined by…

A:

Q: Prove that the function f(r) = is uniformly continuous on (0, 00) エ十2

A:

Q: Solve the Higher order differential Equations. (Choose one only) d*y d?y 4 + 4y = 0 dx2 1- dx4 d*y…

A:

Q: Let f(x)=1/x. Show that f is bounded from above on any interval (α,1), where α>0, but the function…

A:

Q: 1. Let f(x) = x. Show that f is uniformly continuous on any finite closed interval [a, B] (Hint: x-y…

A:

Q: 4. (a) Let I = [a, b] and let f and g be bounded functions on I to R. Suppose that f(x) < g(x) for…

A:

Q: A function is said to be Hoelder a continuous on [a, b] iff |f(x) – f(x')| 0. Show that if a > 1…

A:

Q: Prove that f: R→ R is continuous. 2. Suppose that the function f : [0, 1] → R is continuous and that…

A:

Q: Let f, g to be two continuous functions on [a, b] and f(a) ≥ g(a), f(b) ≤ - Prove that co € [a, b],…

A:

Q: Let f(x) be a given continuous function on a closed interval [a, b]. Use the Extreme Value Theorem…

A: given fx is a continuous function on a closed interval a,b to prove- use extreme value theorem to…