Real Analysis Answers

Using the sequential definition of 
the continuity, prove that the funct-ion, f defined by
f(x)= { 3 , if x is irrational
      { -3 , if x is rational
 is discontinuous at each real numbers.

State Bonnet's mean value theorem for integrals.

Apply it to show that:

5

| ∫ (cos x/x)dx | ≤ 2/3

3

Examine the continuity of the function:

f: [ 1,3]→R defined by :

f(x)= [x]/3x-1 where [x] denotes the greatest integer function

Show that on the curve, y= 3x^2-7x+6, the chord joining the points whose abscissa are x=1 and x=2, parallel to the tangent at the point whose abscissa is x=3/2

Check whether the function f given by:

f(x)= (x-4)^2(x+1)^4 has local maxima and loca

minima

For each of the following statements give the converse, the contrapositive and the negation of the statement.

(i) I take plastic bags when I go shopping.

(ii) x∈Bimpliesx∈/Xorx∈/Y.

(iii) Ifx∈A∩Bthenx∈Aand x∈B.

Let S = {a1, a2, a3, ...an}be a set of test scores. Prove using the the indirect method of proof

that if the average of this set of test scores is greater than 90, then at least one of the scores is greater than 90.

 by mathemarical induction on n that

3^n ≥ 2n^2 + 1 for all n ∈ N

Given the function g : R → R defined by

g (x) = {x-1/2x+4 if x̸=−2 and 1/2 x=−2

Find whether or not f is injective and surjective. Find the inverse of f, if it exists.

Find the infimum and supremum in each of the following sets of real numbers: S = {x| − x2 + 6x − 3 > 0

Let a be the supremum of a set of real numbers and let ε > 0 be any real number. there is at least one x ∈ S such that

a−ε<x≤a

where S is the set with the given supremum

Find the following limits

lim( sqrt2n+1− Sqrt2n)

n→∞

lim3n^3 −n+8/(4n(n−1)(n−2))

n→∞

b) Consider the sequence (an) = (−1)^n − 2n

 Explain whether the sequence is monotone increasing or decreasing, whether it is monotone and if limn→∞(an) = −∞