Real Analysis Answers

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

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]