Real Analysis Answers

Suppose that 𝑓 is differentiable and 𝑓 ′′(𝑎) exists. Prove that 𝑓 ′′(𝑎) = lim ℎ→𝑎 (𝑓(𝑎+ℎ)−2𝑓(𝑎)+𝑓(𝑎−ℎ))/ ℎ^2. Give an example where the above limit exists, but 𝑓 ′′(𝑎) does not exist.

Show that the function defined as 𝑓(𝑥) = { sin 1 𝑥 , 𝑥 ≠ 0 0, 𝑥 = 0 obeys the intermediate value theorem

Let 𝑓: (0,1) → ℝ be a bounded continuous function. Show that 𝑔(𝑥) = 𝑥(1 − 𝑥)𝑓(𝑥) is uniformly continuous. 

State suitable conditions and prove that (𝑓𝑔) ′ = 𝑓𝑔 ′ + 𝑓 ′𝑔.

Evaluate

lim 3nΣr=1 n^2/(4n+r)^3

n→∞

Show whether the following functions are uniformly continuous on the given domain.

1. F(x)=x^3 on [-1,1]

2. F(x)= 2x/2x-1 on [1, infinity]

3. F(x)= sinx/x on (0,1)

4. F(x)= 1/x on (0,1)

1/(3x5)+√3/(5x8)+ √5/(7x11) +... test the convergence

Is there a continuous function f:[0,1]~>[0,1] that is not constant in any nontrivial interval such that f^-1{0} is uncountable?