Real Analysis Answers

State and prove weierstrass M- test

Prove the following result:

A function f that is decreesing on [a,b] is integrable on [a,b].

Let {f_n} be a sequence of functions defined on S .show.that there exists a.function f such that f_n converges to f uniformly on S if and.only if the caught condition is satisfied

Define bounded and unbounded variation. Show that every function which is of bounded variation is bounded

Define uniform convergence of sequence of functions. Give an example

Consider the function f:R² to R defined by

f(x,y) ={ (x²y²)/(x⁴+y²) for (x,y) not equal to zero

0 , for (x,y) =(0,0)

Prove that,

1. f_x(0,0)=f_y(0,0)=0

2. f_x is continuous at (0,0)

3. f_y is.not continuous at (0,0)

About how.much will the function f(x,y) =ln√(x²+y²) change if the point (x,y) is moved from(3,4) a distance 0.1 unit straight toward (3,6)?

Consider f:R² to R defined by f(x,y) =(x+y)/(√2) if x=y and f(x,y) =0 otherwise ,show.that f_x(0,0) =f_y(0,0)=0 and D_uf(0,0)=1 ,where ,u=(1/√2,1/√2) Deduce that f is not differentiable at (0,0)

Let m,n be non negative integers and.let i, j element of N be even . let f:R² to R be defined by f(0,0)=0 and

f(x,y) = (x^myⁿ)/(xⁱ+y^j) for (x,y) not equal to (0,0) .show that f is continuous at (0,0) if and only if ,if m_j+m_i >i_j

Show that a sequence in R² is convergent if and only if it is bounded and all it's convergent subsequences have the same limit