Real Analysis Answers

Prove that every real valued convex function on the closed rectangle in R² is bounded

Show that the function f:R² to R defined by f(x,y)= x²+y² is not uniformly continuous

Let f: R² to R defined by f(x,y) =x²+y² .show that f is differentiable at (x₀,y₀) element. Of R² and find gradient of f at (x₀,y₀).

If f:[a,b] to R is monotonic ,prove that f is of bounded variation on [a,b]?

If f is continuous on [a,b] and f' is bounded in(a,b) prove that f is of bounded variation on [a,b]

Let D₁ and D₂ be subsets of R² and let f₁:D₁ to R and f₂: D₂ to R be continuous functions such that f₁(x,y) =f₂(x,y) for all (x,y) subset of D₁ union D₂ ,let D_i = D₁ union D₂ and let f: D to R be defined by,

f(x,y) ={f₁(x,y) if (x,y) element of D₁,

f₂ (x,y) if (x,y) element of D₂

If D_i is closed for i=1,2. Prove that f is continuous

Given a sequence ((x_n,y_n)) is R² .prove that the following,

1. ((x_n,y_n)) is convergent implies ((x_n,y_n)) is bounded.

2. If ((x_n,y_n)) is a bounded sequence the ((x_n,y_n)) has a convergent subsequence

3. ((x_n,y_n)) is convergent if and only if ((x_n,y_n)) is bounded and every convergent subsequence of ((x_n,y_n)) has the same limit.

4. ((x_n,y_n)) is caught if and only if ((x_n,y_n)) is convergent

Prove that the function f:R² to R defined by f(x,y) =

{ xy/(x²+y²) if (x,y) not equal to (0,0)

0, if (x,y) = (0,0)

Is not continuous at (0,0)

Show that ,f defined on [a,b] is of bounded variation on [a,b] if and only if f can be expressed as the difference of two increasing functions

Show that a polynomial f is of bounded variation in every compact interval [a,b]