Math 447: Real Variables Homework 4

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Homework 4

Math 447: Real Variables

(Exercises were written by Dr. Jason Elliot and are copyrighted by NetMath.)

Exercise 1 Let [x] be the “floor” function; that is, [x] is the unique integer such that [x] ≤ x < [x] + 1 for all x ∈ R.

Let f : R → R be the function

Find all the points where f is continuous. Use the “ε-δ” definition of “continuity” to prove that your answer is correct. Make sure you prove that f is continuous at the values you claim, and that f is discontinuous at all other values.

Exercise 2 Let xn be a sequence of real numbers. Prove that if

lim inf xn = lim sup xn = x ∈ R

then xn converges to x.

Exercise 3 Let (xn) be the sequence

Calculate lim inf xn and lim sup xn and prove your answer is correct.

Exercise 4 Prove that f + = sup{ lim f(xn) | lim f(xn) exists and xn → x }

(follow the argument in the lectures for half of the proof and a hint at the other half).

Exercise 5

(a) Let

Prove that 

(b) Let

Calculate  and prove your answer is correct.