# Question: Define when a function f Rn R is independent

Define when a function f: Rn → R is independent of the first variable and find f1 (a, b) for such f. Which functions are independent of the first variable and also of the second variable?

## Answer to relevant Questions

Let be a continuous real-valued function on the unit circle {x € R2 : |x| =1} such that f (0, 1) = g(1, 0) = 0 and g(- x )= - g(x). Define f: R2→R by F (x) = {|x| . g (x/|x| x ≠0, 0 x = 0. (a) If x € ...Two functions f , g : R → R are equal up to nth order at if lim h → o f(a + h) – g(a + h)/hn =0 (a). Show that f is differentiable at if and only if there is a function g of the form g(x) = a0 + a1 ...Suppose f: Rn →Rn is differentiable and has a differentiable inverse F -1: Rn → Rn. Show that (f-1) I (a) = (fi (f-1(a)))-1.If f: R2→R and D2f =0 and D2f =0, show that f is independent of the second variable. If D1f = D2f =0, show that f inconstant.If f : Rn → R is differentiable and f (0) = 0, prove that there exist gi: Rn → R such that f (x) =Post your question