Real Analysis Answers

Let f:[ 0,π/2] → [-1,1] be a function defined by f(x)= cos 2x . Verify that f satisfies the condition of the inverse function theorem. Hence, what can you conclude about the continuity of f^-1?

Check whether or not the function f, defined on R by

f(x) = { 3x^2sin(1/2x), when x≠0

{ 0 ,when x=0

is derivable on R. If it is, is f' continuous at x=0? If f is not derivable , then define a derivable function on R

Let the function , defined by

f(x)= 1/x+3 ; x∈ [3,∞[

Check whether f is uniformly continuous or not on the interval of definition

I. Show that the set ] -6,8[∩]-8,4[ is a neighborhood of -5.

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

{1,-1,2,-2} is a compact set. True or false with full explanation

A. If (an) is convergent then ∞Σ n=1 an is also convergent true or false with full explanation.

B. The sum of two discontinuous function is always discontinuous function.

True or false with full explanation

Show that L(P₂, f) <=U(P₁, f) where f(x)= 3x + 2 is defined over [1,0] and  P₁ ={0,1/2,3/4,1} and P₂={0,1/4,1/2,3/4,1}

Let a function :f R → R be defined by f(x) =2 if x ∈ Q , 4 if x does not belong to Q. Show that f is not continuous at any x ∈ R.