Real Analysis Answers

Let f be a function of bounded variation on [a, b]. Show that f has at most countably many points of discontinuity in [a, b]. 

r the set of all 2*2 commutative and invertible matrices with real entries Does R from a filed ? prove your assertion . Determine whether R2*2 (R) from order filed or not ? if your answer is no can we define an order such that R 2*2 (R) from an order filed ? prove tour assertion

Every continuous function is differentiable.

True or false with full explanation

-2 is the limit point of the interval ]-3,2[.

True or false with full explanation

Show that series Σ x/(1+n^2.x^2) is uniformly convergent in [k,1] where k>1 but not uniformly convergent in [0,1]

Let f : M → Y be a homeomorphism and O an open subset of M. Explain

concisely in no more than two lines of text why f(O) is an open set.

Let f : M → Y be a homeomorphism and O an open subset of M. Explain

concisely in no more than two lines of text why f(O) is an open set

Let k ≥ 0 and f : M → M a k-Lipschitz function. Let ε > 0. Give the largest

number φ > 0, if any, such that ∀x, y ∈ M, d(x, y) < φ implies d(f(x), d(y)) < ε.