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My attempt

Let $(x, y_1)$ and $(x, y_2)$ be two points in the rectangle $R=[-a, a]\times[-b,b].$ Then

$$|f(x, y_1)- f(x, y_2)|\le|x||\sin y_1 -\sin y_2| + |y_1-y_2|$$

I can't understand how to proceed further; please give me some hints

Anne Bauval
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    Do you mean that it's Lipschitz by $y$? In this case there's a hint: apply Lagrange mean value theorem to $\sin y$ and note that $|\cos y| \le 1.$ Also use the inequality $|x| \le a$. – Botnakov N. Aug 24 '23 at 16:37

1 Answers1

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Your function is differentiable. Thus, checking if your function is Lipschitz is just looking at your derivative and checking that it is bounded. Since your derivative is continuous and the set $[-a,a]\times[-b,b]$ is compact, the derivative is bounded. Thus, your function is Lipschitz.

Eric
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