Real Analysis Answers

Show that the function defined as 𝑓(𝑥) = { 𝑥, 𝑥 ∈ ℚ

𝑥 ² , 𝑥 ∈ ℝ − ℚ

is continuous at 1 and discontinuous at 2.

Measure theory

1)find f+ and f- if fx =cosx+1/2 0<x<2π

2)find the measure of the set{x:sinx>=1/2} for 0<x<2π

 Let f:[a, b] → ℝ be a function and let c ∈ (a, b). If f is of bounded variation on [a, b],

prove that f is bounded on [a, b] and v(f,[a, b] = v(f,[a, c] + v(f,[c,d]).

Find the infimum and supremum of {x ∈ R : x²+2x=3}

The sum of the first n terms of a sequence (Un) n € N^* is given by Sn = n(n+1) / n+2. Find

a) The n^th term of the sequence

b) The sum of the terms from the 6^th to the 31^st term inclusive and exclusive.

Evaluate the limit as x turns to 0

Lim [√(5+x) - √5 / x]

17. Let f: I → R, where I is an open interval containing the point c, and let k ∈ R. Prove the following.

(a) f is differentiable at c with f ′(c) = k iff limh→0 [ f (c + h) – f (c)]/h = k.

*(b) If f is differentiable at c with f ′(c) = k, then limh→ 0 [ f (c + h) – f (c – h)]/2h = k.

(c) If f is differentiable at c with f ′(c) = k, then lim n →∞ n[f (c + 1/n) – f (c)] = k.

(d) Find counterexamples to show that the converses of parts (b) and (c) are not true. 

The book is Steven R. Lay, Analysis with an introduction to proof.