if and only if F-1: R(F) →X is bounded.

Q: Examine the continuity of f(z) = 2-√z+4 Z at z = 0

A: Given: fz=2-z+4z We have to find the continuity of f(z) at z=0.

Q: Prove that there does not exist any function f(x, y) such that

A: To show that there does not exist any function that satisfies the given conditions

Q: Find the linearization L(x) at x= a. f(x) = x+ a = - 1

A:

Q: Let A={x e R|-1sx s1} = B, Show that f:A →B given by f (x)=x|x| is a bijecti on.

A: we have to show that given function is bijective

Q: Show that the limit of the function f(x,y)= (0,0). (x+y)² x² + y² is indeterminate at

A: Limit of the function

Q: Prove that a decreasing function f on [a,b]is integrable.

A:

Q: ɔ that h is a continuous functi (x2 — 2, х < 6 h(x) 3= 7х + a, х 6

A:

Q: 2. Let f : [a, b] R be a bounded function. Prove that if f is continuous at ro € [a, b), then L(|f])…

A: Here we use the property of continuty.

Q: Prove that a polynomial of degree n is uniformly continuous on R if and only if n = 0 or n = 1.

A:

Q: f(z) = Im z is continuous

A: Let us use the convergence criterion to prove that Im z is continuous. Let us assume that  znn=1∞ be…

Q: 2. Show that if f : R → R is continuous on R and f(r) = 0 for all r E Q then f(r) = 0 for all x E R.

A:

Q: Prove that L(f) ≤ 0 ≤ U(f). Given that the bounded function f : [a, b] → R has the property that…

A: Given that the bounded function f : [a, b] → R has the property that f(x) = 0 for all x ∈ [a, b] ∩…

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

A:

Q: 2. (a) Let f be defined for all x, y by f(x, y) x-y - x. Show that f is concave (i) by using Theorem…

A: Given: a) f(x, y) =x - y - x2 Theorem 2.3.1: This theorem says that if f is twice continuously…

Q: Derive a finite difference approximation to of(x,y) + f(x,y) using the points f(x, y), ƒ (x+h, y),…

A: Here we use taylor series to find this.

Q: [0, 4). 4. Prove that f(x) = x² is both uniformly continuous and continuous on

A:

Q: Find the linearization L(x) at x = a. 1 f(x) = - x - a = 1 %3D -

A:

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

A:

Q: Prove that the function : R-{1} →R- {2} defined by 2a f(x)= s bijective. 2 1

A:

Q: 2. Use the Squeeze Theorem to show that f(r) = 2 is in R(0, 3).

A:

Q: Let f(x, y, z) = x²y°z – xyz², and compute: (a)

A: In this question we will use basic of vector  calculus i.e gradient of any function and curl of…

Q: Show that the function f(x, y, 2) = (x² + y² + 2e**)4 is strictly convex on R³.

A:

Q: A function f(x,y) is homogeneous of degree 7 in x and y if and only if, __________.

A: Given problem is solved below.

Q: Prove Theorem (AT )T = A

A: Given: Theorem, ATT=A.

Q: Show that the limit of the function f(x,y) = at (0,0) does not exist. (x+y)² x² + y²

A: Limit of a Function: Like a single-variable function, a multi-variable function can also have limit.…

Q: Q1:- Let T =- where W-N(0, 1) and V~xShow that T2-F(1,r).

A: We know that  The ratio of chi-square divided by its degrees of freedom is called the…

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: If f, g: R → R s.t. f and g are uniformly continuous. and bounded, then show that f.g is also…

A: Let f ,g are uniformly continuous And f and g are bounded

Q: 1. Show (in terms of  ε - δ) that a function f : R3 → R defined by f(x; y; z) = (2x + 3y + 4z) is…

A:

Q: 4. Give an example of a continuous function f : (0, 1) → R that cannot be uniformly approximated by…

A:

Q: Find the bounded อน əx solution of อน 2 + U if U(x, 0) = 6e-3x. at

A: We have ∂u∂x=2∂u∂t+u  The initial condition is ux,0=6e-3x Now, ∂u∂x=2∂u∂t+u can be written as…

Q: If f, g : R R s.t. f and g are uniformly continuous. and bounded, then show that f g is also…

A: A differentiable function f:D→R is said to be uniformly continuous on D if for all ∈&gt;0 , ∃ δ&gt;0…

Q: Exercise 0.4. Assume that g : R → R is a periodic continuous function. Prove that f(x) = 2x + g(x)…

A: Continuity and uniform continuity are two different things. A continuous function may not be…

Q: Suppose that f : R →R is continuous and that its image f(R) is bounded. Prove that there is a…

A:

Q: The function on ℤ defined by f(x)=-x^2-5 is onto. True or False ?

A: The given function is                  f(x) =x2-5   Graph:

Q: 1. Determine whether k(x) is continuous at r = 3, where %3D x + 3, x 2.

A: Given  k(x)=x+3,  x&lt;0x2+1, 0≤x≤22x,   x&gt;2 To check the continuity we can use the graph as…

Q: Let X and Y be t function discrete ran X--1 1, 2. y-4, 9 27 P(x, y)- 0.

A: Two dimensional random variable is said to be discrete when it takes at most a finite number of…

Q: Prove that the limit of xy[(x^2 - y^2)/(x^2 + y^2)] is zero as (x, y) goes to (0,0)

A:

Q: Prove the theorem stating that a E R, if x belongs to the neighborhood V¿(a) for every ɛ > 0, then x…

A: To prove the theorem stating that a∈ℝ, if x belongs to the neighborhood Vεa for every ε&gt;0, then…

Q: Determine the are bounded by: y2-8y+8x=0; and x=0

A: The polynomial y2-8y+8x=0 can be rewritten as x=8y-y28. The function is graphed to understand the…

Q: Prove that f : [−4,2] → R defined by f(x) = 7/x+5 is uniformly continuous on the interval [−4, 2]…

A:

Q: 1. Show (in terms of -5) that a function f: [2, 7] R be defined by f(r) V+1 is uniformly continuous.

A:

Q: 2.23 Let 1 - x2 f(x): if x € Q, if r ¢ Q. Find all points p at which f is continuous. Prove your…

A:

Q: Prove whether a function f(x)=x satisfies the Dirchlet conditions.

A: The function f(x)=x is absolutely integrable over any definite interval. It is bounded in bounded…

Q: Find the linearization L(x) of f(x) at x = a. f(x) = a = 0 4x +3

A: topic - linearization of a function

Q: Find the Wronskian for the set of functions.{x, x2, ex, e−x}

A: Given information: x, x2, ex, e-x The Wronskian is given by the following determinant.…

Q: Discuss the continuity in R3 of the function f(x, y, z) %3D x2+ y2+z2 -1

A: Given: f(x,y,z)=1x2+y2+z2-1.

Q: Find the Wronskian for the set of functions.{1, x, x2, x3}

A: we have to find the wronskian for the set of functions. {1, x, x2, x3}

Q: 4. Show that the function f : X → Y defined by f(X) = X U {a} vX €x is a bijection.

A: