Let ? : R^2 → R be given by ?(?, ?) = √︀ |??|.
a. Show ? is continuous at (0, 0).
b. Show ? does not have a directional derivative at (0, 0) along
(1, 1).
c. Is ? differentiable at (0, 0)?
Let A and B be two non empty bounded subsets of R:
1) Let A +B = { x+y/ x ∈ A and y ∈ B} show that sup(A+B)= sup A
+ sup B
2) For c ≥ 0, let cA= { cx /x ∈ A} show that sup cA = c sup
A
hint:( show c supA is a U.B for cA and show if l < csupA then
l is not U.B)
Let R be the relation on Q defined by a/b R c/d iff ad=bc. Show
that R is an equivalence relation. Describe the elements of the
equivalence class of 2/3.
5. (a) Let f : R \ {−1} → R, f(x) = x+1. Show that f is
injective, but not surjective.
(b) Suppose g : R\{−1} → R\{a} is a function such that g(x) =
x−1, where a ∈ R. Determine x+1
a, show that g is bijective and determine its inverse
function.