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.

Expert Solution

Colby Messinger answered 2 years ago

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

Give an example of a continuous function that is not uniformly continuous. Be specific about the...

Give an example of a continuous function that is not uniformly continuous. Be specific about the domain of the function.

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...

Prove or disprove: Between any n-dimensional vector space V and Rn there is exactly one isomorphism...

Prove or disprove: Between any n-dimensional vector space V and Rn there is exactly one isomorphism T : V → Rn .

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.

4. Prove that for any n∈Z+, An ≤Sn.

Prove that τ(n) < 2 n for any positive integer n. This is a question in...

Prove that τ(n) < 2 n for any positive integer n. This is a question in Number theory

Advanced Calculus 1 Problem 1 If the function f : D → R is uniformly continuous...

Advanced Calculus 1 Problem 1 If the function f : D → R is uniformly continuous and α is any number, show that the function αf : D → R also is uniformly continuous. Problem2 Provethatiff:D→Randg:D→Rareuniformlycontinuousthensois the sum f + g : D → R. Problem 3 Define f (x) = 2x + 1 for all x ∈ R. Prove that f is uniformly continuous. Problem 4 Define f (x) = x3 + 1 for all x ∈ R. Prove...

Show that, in n-dimensional space, any n + 1 vectors are linearly dependent. HINT: Given n+1...

Show that, in n-dimensional space, any n + 1 vectors are linearly dependent. HINT: Given n+1 vectors, where each vector has n components, write out the equations that determine whether these vectors are linearly dependent or not. Show that these equations constitute a system of n linear homogeneous equations with n + 1 unknowns. What do you know about the possible solutions to such a system of equations?

Use induction to prove that for any positive integer n, 8^n - 3^n is a multiple...

Use induction to prove that for any positive integer n, 8^n - 3^n is a multiple of 5.

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