Answer Happy

Let a < b, let D ⊂ R n , let Ω = [a, b] × D, and let f : Ω →R be a function that is of the following form: there is a fixed i ∈{1, . . . , n} and a function g : R 2 → R such that f(t, y) = f(t,y1, . . . , yn) = g(t, yi), (t, y1, . . . , yn) ∈ Ω. Show that ifg(t, yi) is Lipschitz in yi then f(t, y) is Lipschitz in y. Hint:make use of the standard inequality |xi | ≤ |x| for any x = (x1, .. . , xn) in R n .
Problem A8. Let a ≤ b, let D C R¹, let N = [a, b] × D, and let ƒ : N → R be a function that is of the following form: there is a fixed i € {1,...,n} and a function g: R² → R such that f(t, y) = f(t, y₁, 1,9 Yn) = g(t, y₁), (t, y₁, , Yn) EN. Show that if g(t, y;) is Lipschitz in y; then f(t, y) is Lipschitz in y. Hint: make use of the standard inequality |x₁| ≤ |x| for any x = (x1,...,xn) in Rn.