Real Analysis Answers

Discuss the continuity/uniform continuity/Lipchitz continuity and differentiability of the functions |𝑥|, |𝑥| 3 , √𝑥 3 on [−1,1].

Discuss the continuity/uniform continuity/Lipchitz continuity and differentiability of the functions |𝑥|, |𝑥| 3 , √𝑥 3 on [−1,1]

Discuss the continuity/uniform continuity/Lipchitz continuity and differentiability of the functions |𝑥|, |𝑥| 3 , √𝑥 3 on [−1,1].

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

𝑥 ² , 𝑥 ∈ ℝ − ℚ

is continuous at 1 and discontinuous at 2.

State, prove and give an example of one-sided versions of the Sandwich theorem corresponds to 𝑓(𝑥) → ∞ and 𝑓(𝑥) → −∞.