Question

Let f : Rn → Rm be such that the inverse of f(U) is open for...

  1. Let f : Rn → Rm be such that the inverse of f(U) is open for all open sets U ⊆ Rm. Prove that f is continuous on Rn.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let x be a vector in Rn written as a column, and define U = {Ax:A∈Mmn}....
Let x be a vector in Rn written as a column, and define U = {Ax:A∈Mmn}. Show that U is a subspace of Rm.
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.
Let U, V be a pair of subspaces of Rn and U +V the summationspace. Suppose...
Let U, V be a pair of subspaces of Rn and U +V the summationspace. Suppose that U ∩ V = {0}. Prove that from every vector U + V can be written as the sum of a vector from U and a vector from V.
Q1: Let T : Rm → Rn be any linear transformation, and denote by 0m the...
Q1: Let T : Rm → Rn be any linear transformation, and denote by 0m the origin of Rm, that is, this is the vector in Rm with all entries equal to zero. Show that T(0m) = 0n. Q2: Give an example of two linear transformations of the plane such that the matrices that represent the transformations do not commute.
let A be a subset of Rn and let x be a point in Rn. Show...
let A be a subset of Rn and let x be a point in Rn. Show that x is a limit point of A if and only if every open ball about x contains a point of A that is not equal to x
Let u, v, and w be vectors in Rn. Determine which of the following statements are...
Let u, v, and w be vectors in Rn. Determine which of the following statements are always true. (i) If ||u|| = 4, ||v|| = 5, and ?||u + v|| = 8, then u?·?v = 4. (ii) If ||u|| = 2 and ||v|| = 3, ?then |u?·?v| ? 5. (iii) The expression (v?·?w)u is both meaningful and defined. (A) (ii) and (iii) only (B) (ii) only (C) none of them (D) all of them (E) (i) only (F) (i) and...
Consider RN the product of denumerably many copies of R. Define f: R → RN by...
Consider RN the product of denumerably many copies of R. Define f: R → RN by f(t) = (t, t,...). Show that f is continuous if RN has the product topology, but is not continuous if RN has the box topology.
Let V be a subspace of Rn and let T : Rn → Rn be the...
Let V be a subspace of Rn and let T : Rn → Rn be the orthogonal projection onto V . Use geometric arguments to find all eigenvectors and eigenvalues of T . Is T diagonalisable?
Let E be an n×n matrix, and letU= {xE:x∈Rn} (where x∈Rn is written as arow vector)....
Let E be an n×n matrix, and letU= {xE:x∈Rn} (where x∈Rn is written as arow vector). Show that the following are equivalent. (a) E^2 = E = E^T (T means transpose). (b) (u − uE) · (vE) = 0 for all u, v ∈ Rn. (c) projU(v) = vE for all v ∈ Rn.
Let f be a continuous function on the real line. Suppose f is uniformly continuous on...
Let f be a continuous function on the real line. Suppose f is uniformly continuous on the set of all rationals. Prove that f is uniformly continuous on the real line.