Real Analysis Answers

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

Let a,b,c,d element of R with a<b &c<d then show that every real valued convex function on the closed rectangle [a,b]×[c,d] in R² is bounded

State and prove the weierstrass M -test for the uniform convergence of a series of functions

Let D be convex and open in R² and let f:D to R be convex .let [a,b] ×[c,d] be a closed rectangle contained in D ,where a,b,c,d element R with a<b and c<d .prove that there exists k element of R such that ,

|f(x,y)-f(u,v)|<k (|x-u|+|y-v|) for all ((x,y);(u,v)) element of

[a,b]×[c,d].

If a function f of two variables is differentiable, prove that all it's directional derivatives exist and they can be computed by D_uf=del fu

Define absolutely continuous function on [a,b] .suppose f is absolutely continuous, prove that |f| is absolutely continuous.