Real Analysis Answers

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.

Let D is a subset of R² ,S_r(x₀,y₀) subset of D for some r>0 and f:D to R .prove that f is continuous at (x₀,y₀) if and only if if the limit of f as (x,y) tends to (x₀,y₀) exists and is equal to f(x₀,y₀).

Let D is a subset of R² and (x₀,y₀) element of R² be such that D contains S_r(x₀,y₀) \{(x₀,y₀)} for some r>0 and let f:D to R be any function .prove that f(x,y) tends to infinity as (x,y) converges to (x₀,y₀) if and only if the (alpha -delta) condition is true

Let f:R² to R defined by f(x)=√(x²+y²) show that f is continuous on R²

If (a_n) converges to a and (b_n) converges to b show that (a_n+or - b_n) converges to (a+or - b)

Show by an example that in general a continuous function is neither convex nor concave