9.J. If F is a non-empty closed set in R and if x ¢ F, is there a unique point of F that is nearest to x?

Q: Let A be a cut. Show that if r ∈ A and s /∈ A, then r < s. To dispel any suspense, let’s get…

A: Consider the definition for the Dedekind cut A.

Q: What, if any, are the removable discontinuities of f(x) = ? How do you remove them?

A: Give that: The function f(x) = 1x .

Q: Let X, Y be two sets. Prove that X × Y C P(P(X UY )).

A: It is given that X,Y  be two sets. To show that X×Y⊆PPX∪Y Firstly, we define the set X×Y as, X×Y=x,y…

Q: On what interval is In(x² + 1) concave up? くxく

A: This question is about concavity of a function

Q: Suppose that X and Y are topological spaces and f: X- Y. Show that f is continuous iff for each…

A: Definition: Let X and Y are topological spaces and f : X→Y be a function. We say that the function f…

Q: Is the following bounded or unbounded and why? f(x,y) = ln(x2 + y2)

A:

Q: If x and y are elements of an ordered integral domain D,prove the following inequalities:   x2 + y2…

A: An ordered integral domain D is an integral domain with a subset of positive elements with the…

Q: prove the following claims: suppose that ƒ is a convex differentiable function, then for any x, ye R…

A: To prove: Suppose f is a differentiable function then for any x,y∈ℝn, the following inequality…

Q: Consider the set F=(-infinity, 1) U (9,infinity) as a subset of R with d(x,y)=abs(x-y). Is 9 an…

A: Let (M,d) be a metric space and let S⊆M. A point a∈S is said to be an Interior Point of S if there…

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: If F = x? a, – xz a, – y?a,, evaluate f F - dl around the path shown in the diagram

A: Given function is, F=x2ax^-xzay^-y2az^ Then by using Stokes's theorem we have to calculate ∮F·dl→…

Q: Show that x is continuous everywhere. r(x)-x-0 XX<0

A:

Q: 3. Consider the set f = {(x,y) E Z × Z|3x + y = 4}. Is this a function from Z x Z.

A: We need to determine if the given function f=x,y∈ℤ×ℤ3x+y=4 is from ℤ×ℤ Now, 3x+y=4 This means,…

Q: If f is continuous on [a, b], then the horizontal line y = k for every k between f (a) and f (b)…

A: If function 'f'  is continuous on closed interval [a, b], and if f(a)≠ f(b), then for any number k…

Q: With X a topological space, is the function defined as A:X →X × X,x → (x, x) continuous? Investigate

A:

Q: If f is continuous everywhere, then af(bx) dæ < af(bx) dx, where a and b are real numbers. -3 O True…

A:

Q: Let R have its usual topology and let X = {a,b, c}. Define f : R –→ X by а, х

A: Given, ℝ have its usual topology and let X=a, b, c. Define f: ℝ→X   by…

Q: Stafe it fre andwer, your er Let f and be cenfirious on a Space X into R."Then te poduct fxg…

A:

Q: Prove that f (x) = x-12 is : (i) not sectionally continuous in every finite interval in the

A:

Q: Suppose that f:[a,b] R is bounded, P is parition of [a,b] and Q is refinement ofP, then: L(f;P) <…

A: Answer

Q: Verify that the Intermediate Value Theorem applies to the indicated interval and find the calue of c…

A: Given function is

Q: , find the x-values (if any) at which f is not continuous. Which of the discontinuities are…

A: From the given problem: 57).   fx=12x+1,x≤23-x,x&gt;2   As we know that, if L.H.L is not equal to…

Q: ind the values of c that make f continuous everywhere: { Sa² – c² if x 3

A:

Q: Show that x| is continuous everywhere. r(x)- x- 0 xX <0

A: f(x) is continuous, if LHL=RHL=f(x&gt;0), where LHL is left hand limit while RHL is right hand…

Q: O No, because f is not continuous on the closed interval [a, b].

A: Given that: The function f ( x) = x2 - 81x - 4     ,  [ -9, 9  ]  .

Q: Let f(x)= Find f'o. Hint: do not evaluate the in tergral,

A: We will use the Leibnitz formula for differentiation of an integral.

Q: iii If the discontinuity of ƒ is removable, redefine ƒ so that it's continuous at 0

A: This question related to the topic continuity of real analysis.

Q: 7. Show that the function f : R → R* defined by f(x) is not onto. x2 +1

A:

Q: (a) Prove that if m ₹ N, then the function f(x) = xm is continuous on R.

A: We have to prove that, the function f(x) = xm,   m∈ℕ  is continuous on ℝ.

Q: (d) Consider the set [1, √5] C [−1, √10]. Explain f is not continuous by connectedness, compactness,…

A: Given: The function, f:-1,10→ℝ defined by, fx=x2,x&gt;2x,x≤2 The set 1,5⊂-1,10. To explain: Why the…

Q: The set S=xER:x²-4<0} with the usual metric is . O A. Compact. B. connected. C. Not connected D.…

A:

Q: 6. Is the function h: Z → Z defined by ( 2n if n 20 h(n) -n if n < 0 one-to-one? Is it onto?

A: The solution is given as follows

Q: has

A: We have to choose the correct option from the given options according to the given information

Q: 5 Show that the function h(r, y) defined by if r +y? < 1 h(x, y) 0; if r +y 2 1 is discontinuous.

A: For continuity of 2 variable function L.H.L=R.H.L=Function value at (x,y). limx→a+,y→b+…

Q: Let f: R → R with the usual topology that sends compact sets to compact sets in R ^ 2 Is the…

A: For any x,y∈R if y=fx there there exists a number x±∈R in neighborhood of x such that y±=fx±.

Q: On which of the following interval is f(x) S O (0, +∞0) ○ (2, +∞) O [2, +∞0) 0 (-∞, +∞0) 1 x-2…

A: We will the fact that root value should be positive.

Q: 2. Show by definition that the function f : R R, f(x) = ||, is continuous on R. Hint: you may need…

A:

Q: Let f = f(x, y), x = s +t and y = s - t. Show that (9- ) - se se as at ax ду

A: It is given that, f=f(x,y), x=s+t and y=s-t. Show that, ∂f∂x2−∂f∂y2=∂f∂s·∂f∂t.…

Q: Suppose f is a bounded real function on [a, b), and f' eR on [a, b). Does it follow that fe R? Does…

A:

Q: جليلة Let I = [= 1,₂1] ISR interval w that I is negligible Set.

A:

Q: Suppose A is an uncountable set and B is a denumerable set. Suppose f:A→B is a function. Prove that…

A:

Q: Suppose h is a function such that h(1) = -2, h'(1) = 2, h"(1) = 3, h(2) = 6, h'(2) = 5, h"(2) = 13,…

A: Given: The second derivative is continuous everywhere.

Q: 20. Give two examples of functions from Z to Z that are one- to-one but not onto.

A:

Q: Let S be some non-empty collection of real numbers. We know that Vx E S, 3y E S such that x2 y.…

A: In this type of question you need to assume the set of x,y and S that satisfy the condition given in…

Q: Is the set θ={((x,y),(3y,2x,x+y)) : x,y∈R} a function? If so, what is its domain and range? What can…

A:

Q: Show that the following statement is false by drawing a graph that provides a counterexample: If f…

A:

Q: If f:R → R is continuous at both 2 and 4, then f must be continuous at some c E (2,4). Select one:…

A: Consider the function f:R→R, that is continuous at both the points 2 and 4. Let there be a point c…