Real Analysis Answers

For each x ∈ R, let us denote by C(x) the least integer greater than or equal to x.
For example, C(1) = 1, C(−√2) = −1. In other words, C(x) is the unique integer
satisfying C(x) − 1 < x ≤ C(x).
(1) Draw the graph of the function C(x) for x ∈ [−2, 2].
(2) Prove that C(x) is continuous at all non-integer points of R.
(3) Prove that C(x) is discontinuous at all integer points of R.

Prove that the sequence {f_n(x)} where f_n(x)=nx/1+n^2x^2
is not uniformly convergent in [-1,1].

Show that the function f:[2,3]→R defined by :
f(x)={0 if x is rational {1 if x is irrational
is discontinues and not integrable over [2,3].Does it imply that every discontinues function is non-integrable?Justify your answer.

Obtain the value of x for which the series Σ 1.3.5.......(4n-3)/2.4.6.........(4n-2) x^(2n)/4n(>0) is convergent.

Are the following statements true or false?Give reasons for your answers and explain in details.
(a)Every subsequence of the sequence {1\n^(2)} is convergent
(b)The function f(x)=x^2+|x|,is differentiable at x=-1
(c)Every infinite set is an open set.
(d)A necessary condition for a function f to be integrable is that it is continuous.
(e)The sum of the series 3n∑r=1 1/3n+2r as n→∞ can be calculated by evaluating the integral ∫ 1/3+2x dx from 0 to 3.

1) EVALUATE

Limit {(2+x)^5/4-(2)^5/4}/{(2+x)^2/3-(2)^2/3}
x tends to zero

Find whether the following sequences converge or not


A) {2+(-1)^n}

B)(4n^3+n)/(2n^3+7n)

a) Every infinite set is an open set.

b)A necessary condition for a function f to be integrable is that it is continuous.


true or false

The function f (x) = x2 + x , is differentiable at x = −1

Every subsequence of the sequence


(1/n^2)
is convergent