Real Analysis Answers

Check whether the interval [7,10[ and ]3,6] are equivalent or not

If (an) is convergent then "\\sum_{i=1}^{\\infty} a_n" is also convergent. True or false with the full explanation.

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