Real Analysis Answers

Consider the metric space (B(s),d) of all bounded real valued functions on a non empty set S , with metric d(f,g) =||f-g|| ,where ||f|| =sup _{x element of s}{|f(x)|} prove that f_n converges to f in the metric space (B(s),d) if and only if f_n converges to f uniformly on S

Let D is a subset of R² ,(x₀,y₀) element of D and let f: D to R be any function, prove that the following are equivalent.

1. f is continuous at (x₀,y₀).

2. For every epsilon > 0 ,there is a delta >0 such that |f(x,y) -f(x_0,y₀)| epsilon for all (x,y) element. Of D union S_delta (x_0,y₀)

3. For every open subset V of R containing f(x_0,y₀) there is an open subset U of R² contain (x₀,y₀) such that f(U union D) subset of V

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]

Let f be of bounded variation on [a,b] and define V is,

V(x)= {0 if x=a

V₁(a,x) if a less than or equal to x.less than or equal to b

Then show that a point of continuity of f is also a point of continuity of V and conversely

Show that a function of bounded variation on [a,b] is bounded therein

Let alpha be of bounded variation on [a,b] and assume that f element of R(alpha) on [a,b] .show that f element of R(alpha) on every subinterval [c,d] of [a,b]