1 Define the function of by cos(27") 3n f(x) = Σ n=0 a) Use the Weierstrass M-test to show that f is continuous for all 2 € R. b) Show that f is uniformly continuous on all of R.

Q: (1) Define the function f : RR by f(x) at a = 1. 2 X x2 x <1 x≥1° Show that f is continuous

A: অতএব, f a=1 এ অবিচ্ছিন্ন।Explanation:

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: Define f : Rm \ {0} → R by ƒ(x) = T: Prove that f is uniformly continuous on R" \ B1(0), but not…

A:

Q: Question 5 Given matrices A and D 0 0 1 A =0 1 0 10 0 1 D =0 -20 Which of the following is correct?…

A: It is given that ;                   A = 001010100      and    D = 1000-20001 And we have to find…

Q: 7.) . moments of f equal the moments of Let f be a continuous function on [-1, 1] with the property…

A: Given information: F∈-1,1gt=11+t2

Q: -1+x+ 4x² cos (4) når r + 0 når x = 0. f(x) = %3D %3D

A: We have given function;

Q: 2. Determine if a function f : [0 ; 2pie] → R defined by f(x) = sin 2pie is continuous

A: Determine if a function f : [0 ; 2pie] → R defined by f(x) = sin 2pie is continuous.

Q: Consider the Cobb-Douglas function f (x1,x2,..,Xn) = x"x i=1 where x; > 0, and a; > 0 for all…

A:

Q: The function f is a composite of two continuous functions: y = cos x and y = x 2, so f is continuous…

A: The function fx is a composition of functions y=cosx and y=x2and it is a continuous function.

Q: Prove or disprove: The function f : [0, 1] → R defined by x sin (¹) if 0 < x≤1, {₁ 0 if x = 0, f(x)…

A: Theorem: If f : K→R is continuous and K⊂R is compact, then f is uniformly continuous on K. Given:…

Q: 6. (a) Use the Schröder-Bernstein Theorem to prove that [a, b]| = |(a,b)| for any a, b e R. (b) Use…

A:

Q: x² – 16 tate why f(x) is continuous on [1, 2]. x² – 9

A: f(x) is continuous on [a,b] if and only if  1. ∀x0∈(a,b)⊆R, limx→x0f(x)=f(x0) 2. limx→a+f(x)=f(a)…

Q: Let f(x) = ri- 2xi.Find the value of c that satisfies the conclusion of Rolle's Theorem forf on (0,…

A: The graph is shown below:

Q: 3. Let fn(x) = ¹+cos(nx), x € R. Find f(x) so that fn → ƒ pointwise on R, then check whether fn → f…

A:

Q: Let f(t) = t for t > 0, and g(t) = sin t for t > 0. Compute f * g.

A: If we are taking * as the product, then f*g=fg=t sin t for t≥0. Taking * as convolution: We know…

Q: Let f be the function defined by: sin x, x 2 - For what values of x is f NOT continuous?

A: Given function is: fx=sinxx&lt;0x20≤x&lt;12−x1≤x&lt;2x−3x≥2

Q: Suppose that f :C → C is integer and that ƒof = f. What can you say about f ? Justify your answer.…

A: Please find the answer in next step

Q: Let = [ " - 1)5(1 – 2° drt a) find the intervals on which f is increasing and on which f is…

A:

Q: If that f is continuous on a sxsb. Let F(x) be any of f(x). Then, f(x)dx = F(b) - F(a). This theorem…

A: Recall: Mean value theorem: If f(x) is continuous over an interval [a,b], then there is atleast one…

Q: Assume that f is continuous on the set of real numbers. Find a and b given that b 3 [ f(x) dx - f₁…

A:

Q: Let f be the function given by ƒ (z)=5 cos² () + In(z + 1) – 3 The derivative of f is given by f'…

A:

Q: Which of the following functions is continuous on the interval (-0, +)? …* f(x) = Vx² – 1 (A f(x) =…

A:

Q: Let the function ƒ : [0, 1] →R be defined by { if x = ! (n = 1, 2, 3, · · · ) f(x) = n 1– x2…

A: Given :  f : [0, 1] →ℝ  defined by ;              fx = ex2,if  x=1n    (n=1, 2, 3, ...…

Q: c) Consider a function f defined on the real numbers as { 0ifais irrational. S(r) = 1/n ifr= m/n…

A: To show that the function f is continuous at all irrational numbers.

Q: Consider a function ƒ : [0, 1] → R defined by f(x) = x(x³ - 1). (a) Can we apply Rolle's Theorem to…

A: