Real Analysis Answers

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.

Consider any two positive real numbers and call it

Let d₁ and d₂ be two metrics for the set X and suppose that there is a positive number c such that d₁(x,y) less than or equal to cd₂ (x,y) for all x,y element of X .Then prove that the identity function , (X,d₁) converges to (X,d₂) is continuous

 Consider any non-zero point in 𝑅2 and name it (𝑎,𝑏). Then                 

(i). Write any four different paths that passes through your chosen point (𝑎,𝑏).

(ii). Compute the limits of the following function when (𝑥,𝑦) → (𝑎,𝑏) along all these four paths,                            

 𝑓(𝑥,𝑦) = {

(𝑥−𝑎)2(𝑦−𝑏) (𝑥−𝑎)4+(𝑦−𝑏)2

, (𝑥,𝑦) ≠ (𝑎,𝑏) 0, (𝑥,𝑦) = (𝑎,𝑏)  

(iii). Conclude from the results obtained in (ii) and answer whether lim (𝑥,𝑦)→(𝑎,𝑏) 𝑓(𝑥,𝑦) exists or not.  

(iv). Is the function 𝑓(𝑥,𝑦) continuous at the origin? Explain.  

(v). Also calculate 𝑓𝑥(𝑎,𝑏) and 𝑓𝑦(𝑎,𝑏).  

(vi). Write all points of differentiability of 𝑓.

 Consider any two positive real numbers and call it 𝑎 and 𝑏. Then consider the function defined as  

𝑓(𝑥) = {

𝑎, 0 ≤ 𝑥 < 1 𝑏,          𝑥 = 1 Find the 𝑈(𝑃,𝑓) and 𝐿(𝑃,𝑓) for the partition 𝑃 = {𝑥0,𝑥1,…𝑥𝑛} of [0,1]. Also check whether the function is Riemann integrable over [0,1] or not. 

Let a,b,x be three real numbers with a>b and x>0. Which of the following statements is correct?

A. X^a>X^b if a,b>1 and for every x>0

B. X^a<X^b if X is an element of (0,1)

C.X^a<X^b if a, b>0 and for every x>1

D.X^a>X^b if a,b x>0