Real Analysis Answers

Consider the function

F(x)= xsin(π/x) 0<x≤1

0 x=0

Show that f(x) is continuous but not of bounded variation

What is the limits of (1/n)

Test whether the series ∞Σn=0 1/(n^5+x^3) converges uniformly or not

Show that the function f defined on [0,1] by f(x)= (-1)^(n-1) for 1/(n+1) < x/n ≤ 1/n where (n=1,2,3...) is integrable on [0,1]

Find whether the following series are convergent or not

i. ∞Σ n=1 (3n-1)/7^n

Let f be a function defined on R by:

F(x)= { x+5/(x^2-25) , when x≠5

{ 1 when x= 5

Check whether f is uniformly continuous on [-3,3] or not

Show that the function f: [0,1]→ R defined by

F(x) = { 2 , when x is rational

{ 3 , when x is irrational

is not riemann integrable on[0,1]

Give an example of each of the following:

i. a function with a removable discontinuity

ii. a totally discontinuous function

Check whether the sequence {an}, where

an= 1/(n+1)+1/(n+2)+..1/2n is convergent or not

The function f: R→R defined by f(x)= | x-1|+ | 3-x| is differentiable at x= 4.

True or false with full explanation