Question

Suppose ? ⊂ R^? , ? ⊂ R^? are nonempty and open and ? : ?...

Suppose ? ⊂ R^? , ? ⊂ R^? are nonempty and open and ? : ? → R^? and ? : ? → R^? . Let ℎ : ? × ? → R ?+? be defined by ℎ(u, v) = (?(u), ?(v)). If ? is continuous at x ∈ ? and ? is continuous at y ∈ ? , then show that ℎ is continuous at (x, y) ∈ ? × ? .

Hint: Note that for any vectors z ∈ R ? and w ∈ R ? , ‖(z, w)‖ = ‖(z, 0) + (0, w)‖ ≤ ‖(z, 0)‖ + ‖(0, w)‖ = ‖z‖ + ‖w‖. And, for any vectors u ∈ R ? and v ∈ R ? , ‖(u, v)‖ ≥ max (‖u‖, ‖v‖).

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let A be open and nonempty and f : A → R. Prove that f is...
Let A be open and nonempty and f : A → R. Prove that f is continuous at a if and only if f is both upper and lower semicontinuous at a.
a)Suppose U is a nonempty subset of the vector space V over field F. Prove that...
a)Suppose U is a nonempty subset of the vector space V over field F. Prove that U is a subspace if and only if cv + w ∈ U for any c ∈ F and any v, w ∈ U b)Give an example to show that the union of two subspaces of V is not necessarily a subspace.
Show that GL(2, R) acts transitively on R^2 − {0}. Hint: Let v, w ∈ R^2...
Show that GL(2, R) acts transitively on R^2 − {0}. Hint: Let v, w ∈ R^2 be any nonzero vectors and consider polar coordinates.
3. a. Consider R^2 with the Euclidean inner product (i.e. dot product). Let v = (x1,...
3. a. Consider R^2 with the Euclidean inner product (i.e. dot product). Let v = (x1, x2) ? R^2. Show that (x2, ?x1) is orthogonal to v. b. Find all vectors (x, y, z) ? R^3 that are orthogonal (with the Euclidean inner product, i.e. dot product) to both (1, 3, ?2) and (2, 7, 5). C.Let V be an inner product space. Suppose u is orthogonal to both v and w. Prove that for any scalars c and d,...
Suppose K is a nonempty compact subset of a metric space X and x∈X. Show, there...
Suppose K is a nonempty compact subset of a metric space X and x∈X. Show, there is a nearest point p∈K to x; that is, there is a point p∈K such that, for all other q∈K, d(p,x)≤d(q,x). [Suggestion: As a start, let S={d(x,y):y∈K} and show there is a sequence (qn) from K such that the numerical sequence (d(x,qn)) converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}. Show, there is a point z∈X and distinct points a,b∈T that are nearest points to...
Suppose is orthogonal to vectors and . Show that is orthogonal to every in the span...
Suppose is orthogonal to vectors and . Show that is orthogonal to every in the span {u,v}. [Hint: An Arbitrary w in Span {u,v} has the form w=c1u+c2v. Show that y is orthogonal to such a vector w.] What theorem can I use for this?
Suppose A is a subset of R (real numbers) sucks that both infA and supA exists....
Suppose A is a subset of R (real numbers) sucks that both infA and supA exists. Define -A={-a: a in A}. Prive that: A. inf(-A) and sup(-A) exist B. inf(-A)= -supA and sup(-A)= -infA NOTE: supA=u defined by: (u is least upper bound of A) for all x in A, x <= u, AND if u' is an upper bound of A, then u <= u' infA=v defined by: (v is greatest lower bound of A) for all y in...
1. Let W be the set of all [x y z}^t in R^3 such that xyz...
1. Let W be the set of all [x y z}^t in R^3 such that xyz = 0. Is W a subspace of R^3? 2. Let C^0 (R) denote the space of all continuous real-valued functions f(x) of x in R. Let W be the set of all continuous functions f(x) such that f(1) = 0. Is W a subspace of C^0(R)?
Suppose S ⊂ R is nonempty and M is an upper bound for S. Show M...
Suppose S ⊂ R is nonempty and M is an upper bound for S. Show M = sup S if and only if for every Ɛ > 0, there exists x ∈ S so that x > M − Ɛ.
For a nonempty subset S of a vector space V , define span(S) as the set...
For a nonempty subset S of a vector space V , define span(S) as the set of all linear combinations of vectors in S. (a) Prove that span(S) is a subspace of V . (b) Prove that span(S) is the intersection of all subspaces that contain S, and con- clude that span(S) is the smallest subspace containing S. Hint: let W be the intersection of all subspaces containing S and show W = span(S). (c) What is the smallest subspace...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT