Prove that a continuous function for sections(subrectangles) is integrable in R^N

Expert Solution

Colby Messinger answered 2 years ago

Let f : R → R be a function. (a) Prove that f is continuous on...

Let f : R → R be a function. (a) Prove that f is continuous on R if and only if, for every open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is open. (b) Use part (a) to prove that if f is continuous on R, its zero set Z(f) = {x ∈ R : f(x) = 0} is closed.

Prove that if f, g are integrable, then the function (f(x) + cos(x)g(x))2is integrable.

Prove that any continuous function on a standard n-dimensional cube In can be uniformly approximated by...

Prove that any continuous function on a standard n-dimensional cube In can be uniformly approximated by polynomials in n variables x1, . . . , xn.

Consider a continuous, integrable, twice-differentiable function f with input variable x. In terms of the units...

Consider a continuous, integrable, twice-differentiable function f with input variable x. In terms of the units of f and the units of x, choose the units of each function or expression below: (a) The units of f ' are the units of f the units of x (the units of f)(the units of x ) the units of f the units of x the units of f (the units of x)2 the units of f (the units...

Prove 1. For each u ∈ R n there is a v ∈ R n such...

Prove 1. For each u ∈ R n there is a v ∈ R n such that u + v= 0 2. For all u, v ∈ R n and a ∈ R, a(u + v) = au + av 3. For all u ∈ R n and a, b ∈ R, (a + b)u = au + bu 4. For all u ∈ R n , 1u=u

Let φ : R −→ R be a continuous function and X a subset of R...

Let φ : R −→ R be a continuous function and X a subset of R with closure X' such that φ(x) = 1 for any x ∈ X. Prove that φ(x) = 1 for all x ∈ X.'

1) There is a continuous function from [1, 4] to R that is not uniformly continuous....

1) There is a continuous function from [1, 4] to R that is not uniformly continuous. True or False and justify your answer. 2) Suppose f : D : →R be a function that satisfies the following condition: There exists a real number C ≥ 0 such that |f(u) − f(v)| ≤ C|u − v| for all u, v ∈ D. Prove that f is uniformly continuous on D Definition of uniformly continuous: A function f: D→R is called uniformly...

True and False (No need to solve). 1. Every bounded continuous function is integrable. 2. f(x)=|x|...

True and False (No need to solve). 1. Every bounded continuous function is integrable. 2. f(x)=|x| is not integrable in [-1, 1] because the function f is not differentiable at x=0. 3. You can always use a bisection algorithm to find a root of a continuous function. 4. Bisection algorithm is based on the fact that If f is a continuous function and f(x1) and f(x2) have opposite signs, then the function f has a root in the interval (x1,...

Exercise 2.5.1 Suppose T : R n ? R n is a linear transformation. Prove that...

Exercise 2.5.1 Suppose T : R n ? R n is a linear transformation. Prove that T is an isometry if and only if T(v) · T(w) = v · w. Recall that an isometry is a bijection that preserves distance.

a) Let S ⊂ R, assuming that f : S → R is a continuous function,...

a) Let S ⊂ R, assuming that f : S → R is a continuous function, if the image set {f(x); x ∈ S} is unbounded prove that S is unbounded. b) Let f : [0, 100] → R be a continuous function such that f(0) = f(2), f(98) = f(100) and the function g(x) := f(x+ 1)−f(x) is equal to zero in at most two points of the interval [0, 100]. Prove that (f(50) − f(49))(f(25) − f(24)) >...

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