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.

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]