Introduction to Analysis
Homework 14
1. (Rudin ex9.7) Suppose that f is a real-valued function defined in an open set E ⊂ Rn, and that the partial derivatives D1f, · · · , Dnf are bounded in E. Prove that f is continuous in E.
2. (Rudin ex9.15) Define f (0, 0) = 0, and put
f (x, y) = x2+ y2− 2x2y − 4x6y2 (x4+ y2)2 (a) Prove, for all (x, y) ∈ R2, that
4x4y2 ≤ (x4 + y2)2. Conclude that f is continuous.
(b) For 0 ≤ θ ≤ 2π, −∞ < t < ∞, define
gθ(t) = f (t cos θ, t sin θ).
Show that gθ(0) = 0, gθ0(0) = 0, gθ00(0) = 2. Each gθ has therefore a strict local minimum at t = 0.
In other words, the restriction of f to each line through (0, 0) has a strict local minimum at (0, 0).
(c) Show that (0, 0) is nevertheless not a local minimum for f , since f (x, x2) = −x4. 3. Let I = [a − 1, a + 1] and f : R → R be a differentiable function with f (a) = b and
1 < |f0(x)| < 3 for x ∈ I. Let y ∈ (b − 1, b + 1).
(a) Use the idea of Newton’s method to prove that there is x ∈ I such that f (x) = y.
Let x0 = a and
xn+1 = xn+ 1
2(y − f (xn)).
Prove that there is ¯x such that xn → ¯x as n → ∞. Hence, y = f (¯x).
(b) Use Fixed Point Theorem to prove that there is x ∈ I such that f (x) = y. Define φ(x) = x + 1
2(y − f (x)).
Prove that φ(x) is a contraction mapping from I into I.
4. Let f(x, y) = (2x2 + 10y + xy, x3 + y3 + 20x). Prove that f 0(x, y) is invertible for all
−1 ≤ x, y ≤ 1.
5. Let L : Rn → Rn be a linear isomorphism and f (x) = L(x) + g(x), where |g(x)| ≤ M |x|2 and f is C1. Show that f is invertible near 0. (We say that h : Rn→ Rnis an isomorphism if |h(x) − h(y)| = |x − y|.)
6. Let f(x, y) = (excos y, exsin y). Prove that f 0(x, y) is invertible for all (x, y) ∈ R2 but f is a not one-to-one function.
