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Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be continuously differentiable. Suppose $|f_x (x,y) | \leq K$, $|f_y (x,y) | \leq K$ for all $(x,y)$. Prove that $$|f(x_1, y_1) - f(x_2, y_2)| \leq \sqrt 2 K \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$
My approaches don't seem to work here. Give me an idea!

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    Try using the fundamental theorem of calculus in the following way: $|f_x(x,y)| \leq K$ means that $f(x_1,y_1) - f(x_2,y_1) = \int_{x_2}^{x_1} f_x(s,y) ds \leq \int_{x_2}^{x_1} K ds$ and the same for $f_y(x,y)$.2011-12-12

2 Answers 2