Real Analysis Answers

The sum of two discontinuous function is always discontinuous function. True or false with full explanation

Every integrable function is monotonic. True or false with full explanation.

Every continuous function is differentiable.

True or false with full explanation.

Check whether the sequence (an), where

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

Let ϕ and Ψ be function defined on [-3,5], such that both are continuous on [-3,5], derivable in [-3,5] and ϕ'(x)= Ψ'(x) ∀ x∈]-3,5[. Prove that

ϕ(x)= Ψ(x) +c ∀ x∈ [-3,5] , where c is a real constant

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

True or false with full explanation

i. Every continuous function is differentiable.

ii. Every integrable function is monotonic

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

Examine the following series for convergence: "\\sum _{n=0}^{\\infty }\\left(\\frac{n-2}{2n+3}\\right)^n"

find "\\lim _{n\\to \\infty }\\left(\\frac{1}{\\left(2n+1\\right)^2}+\\frac{2}{\\left(2n+2\\right)^2}\\frac{3}{\\left(2n+3\\right)^2}\\right)+...+\\frac{3}{25n}"