Let \(F:\RR^2\to \RR\) be continuously differentiable with \(F(0, 0) = 0\) and \(\norm{\nabla F(0, 0)} < 1\).
Prove that there is some real number \(r> 0\) such that \(\abs{F(x, y)} < r\) whenever \(\norm{(x, y)} < r\).
Solution
1. Choose a constant \(c \in \mathbb{R}\) such that \(\|\nabla F(0, 0)\| < c < 1\).
\(\|\nabla F(0, 0)\| < 1\) by hypothesis, so \(c = \frac{\|\nabla F(0, 0)\| + 1}{2}\) satisfies the inequality.
2. There exists \(r > 0\) such that \(\|\nabla F(u)\| \le c\) for all \(u \in \mathbb{R}^2\) with \(\|u\| \le r\).
2.1. The function \(F\) is \(C^1\), so the gradient mapping \(\nabla F: \mathbb{R}^2 \to \mathbb{R}^2\) is continuous.
definition of continuously differentiable function.
2.2. The Euclidean norm \(\|\cdot\|: \mathbb{R}^2 \to \mathbb{R}\) is continuous, so \(u \mapsto \|\nabla F(u)\|\) is continuous on \(\mathbb{R}^2\).
composition of continuous functions.
2.3. Since \(\|\nabla F(0, 0)\| < c\), the preimage \(\{u \in \mathbb{R}^2 : \|\nabla F(u)\| < c\}\) is an open neighborhood of \((0, 0)\).
preimage of the open interval \((-\infty, c)\) under a continuous function.
2.4. Hence there exists a radius \(r > 0\) such that the closed ball \(\bar{B}_r(0, 0) = \{u \in \mathbb{R}^2 : \|u\| \le r\}\) is contained in this neighborhood.
definition of an open set in \(\mathbb{R}^2\).
3. For any \((x, y) \in \mathbb{R}^2\) with \(\|(x, y)\| < r\), \(|F(x, y)| < r\).
3.1. Fix \((x, y) \neq (0, 0)\) with \(\|(x, y)\| < r\) (for \((x, y) = (0, 0)\), \(|F(0, 0)| = 0 < r\)).
case distinction.
3.2. Define the single-variable function \(g: [0, 1] \to \mathbb{R}\) by \(g(t) = F(t x, t y)\).
restriction of \(F\) to the line segment from \((0, 0)\) to \((x, y)\).
3.3. \(g\) is differentiable on \([0, 1]\) with derivative \(g'(t) = \nabla F(t x, t y) \cdot (x, y)\) by the chain rule.
multivariable chain rule.
3.4. By the single-variable Mean Value Theorem, there exists \(t_0 \in (0, 1)\) such that:\[ F(x, y) - F(0, 0) = g(1) - g(0) = g'(t_0) = \nabla F(t_0 x, t_0 y) \cdot (x, y). \]
Mean Value Theorem for \(g\) on \([0, 1]\).
3.5. The intermediate point \(u_0 = (t_0 x, t_0 y)\) satisfies \(\|u_0\| = t_0 \|(x, y)\| < \|(x, y)\| < r\).
\(t_0 \in (0, 1)\) and \(\|(x, y)\| < r\).
3.6. By 2, \(\|\nabla F(u_0)\| \le c\).
\(\|u_0\| < r\).
3.7. Applying the Cauchy–Schwarz inequality to the dot product gives:\[ |F(x, y)| = |F(x, y) - F(0, 0)| = |\nabla F(u_0) \cdot (x, y)| \le \|\nabla F(u_0)\| \|(x, y)\| \le c \|(x, y)\|. \]
Cauchy–Schwarz inequality in \(\mathbb{R}^2\) and \(F(0, 0) = 0\).
3.8. Since \(c < 1\) and \(\|(x, y)\| < r\), we have:\[ |F(x, y)| \le c \|(x, y)\| < c r < r. \]
\(c < 1\) and \(r > 0\).
4. Conclusion: There exists \(r > 0\) such that \(|F(x, y)| < r\) whenever \(\|(x, y)\| < r\).
2 and 3.
Q.E.D.