Suppose that f : rn → rm and that a ∈ k, where k is a connected subset of rn . suppose further that for each x ∈ k there exists a δx > 0 such that f(x) = f(y) for all y ∈ bδx (x). prove that f is constant on k; that is, f(x) = f(a) for all x ∈ k

True or false when elements combine to form a mixture, the resulting chemical properties may be very different from those of the elements that make it.

Agraduated cylinder contains 63.0 ml of water. a piece of gold, which has a density of 19.3 g/ cm3, is added to the water and the volume goes up to 64.5 ml. calculate the mass in grams of the gold that was added to the water. explain how you got your answer.

Sawyer is studying diffraction. he draws a diagram of a plane wave to show how light waves travel. which best describes sawyer’s error? the wave fronts should be perpendicular to the direction in which the waves move. the arrow showing the direction of movement of the waves should be pointing to the left. the arrow showing the direction of movement of the waves should be pointing downward. the wave fronts should be both parallel and perpendicular to the direction in which the waves move.