1. Let X be G-measurable and let Y be independent of G. Let f(x, y) be a bounded continuous function and define g(x) = E[f(x,Y)]. Show that E[f (X,Y)| G] = g(X). prove it when X is discrete and X is not discrete.

Q: Let (X,F) be a measurable space and let f, g: X → Y be F -measurable functions. If E E F, show that…

A: The given in this question is a measurable space (X,F) and two F-measurable functions f and g from X…

Q: Problem (3) Show that if a continuous function f is holomorphic outside a line in C then it 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: Let f [0, 2] → R be a continuous function, with ƒ(0) ≤ f(1) ≤ f(2). Show that there : exists xo [0,…

A:

Q: Suppose f: [-1,0] → R and g: [0, 1] → R are continuous and f(0) - g(0). Define h: [-1,1] → R by…

A:

Q: 7. Suppose f is a continuous function on (a, b) with b> a. Stat f (recall this theorem was taught in…

A:

Q: A continuous function ƒ : [0, 1] → R satisfies ƒ(0) f(1). Show that for each integer n ≥ 1 there…

A: Let fix δ=1n, n∈ℕ Consider the continuous function gnx:0,1-δ→ℝ; gnx:=fx+1n-fx and suppose that gnx≠0…

Q: Show that the function f: C → C given by f(z) = z2 is continuous on C.

A:

Q: "Prove that the function f : R→ R with ƒ(x) = = is uniformly continuous."

A:

Q: Q. (a) Find an equation of the tangent plane to each surface at the given point. (b) Find symmetric…

A:

Q: 2. Let f(x) = V4 – x for x < 4 and g(x) = x² for all x E R. (a) Give the domain of f · g, ƒ + g, ƒ º…

A:

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

A:

Q: Suppose f : R → R is a continuous function with the property that f(x) = 0 for every x E Q. Prove…

A:

Q: If f:X - Y is a continuous function then f- (V) is open in X for every open set V in Y. True False

A:

Q: Let a < b be given. Give examples of continuous functions g, h : (a, b) → R such that (a) g is not…

A: Let a&lt;b be given.We need find the examples of continuous functions g,h:(a,b)→R such that:(a) g is…

Q: Suppose f: (0,00) → (0, ∞) and g: (0, ∞o) → (0, ∞o) are continuous functions. True or false: If g…

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: continuous function h(x),

A: Differential equation

Q: Let ƒ : R → R be a function that is differentiable on (a, ∞), where a is any real onstant. Let g: RR…

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: B. Let f: R²R be a continuous function, and let S be the subset of the R2 given by all pairs (x, y)…

A: Given: Let fx,y=x2+xy+3y2-3. To Show: fx,y contain minimum and maximum value.

Q: 1. Construct a continuous function f: I R on a bounded interval I with the following property: f is…

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…

Q: 1. If f:(0, 3) → R is continuous, must its range be an open (possibly infinite) interval?

A: Given statement is : If f:(0, 3) → R is continuous, must its range be an open (possibly infinite)…