The function \( f(x, y)=\left\{\begin{array}{ll}\frac{\cos x \sin y}{2 y}, & y \neq 0, \\ 2 \cos x, & y=0\end{array}\right. \) is continuous on the domain

To determine the continuity of \( f(x, y) \) at the point where \( y = 0 \), we first observe that for \( y \neq 0 \), the function is defined as \( \frac{\cos x \sin y}{2y} \). As \( y \) approaches 0, we need to evaluate the limit: \[ \lim_{(x, y) \to (x_0, 0)} f(x, y) = \lim_{y \to 0} \frac{\cos x \sin y}{2y} = \frac{\cos x}{2} \text{ (by using L'Hôpital's Rule)}. \] This limit must equal \( f(x_0, 0) = 2\cos x_0 \) for continuity. Therefore, setting these two expressions equal gives us the value of \( x_0 \) where the function is continuous. To satisfy \( \frac{\cos x_0}{2} = 2 \cos x_0 \), we determine that we need non-degenerate values for \( x_0 \) where this equation holds, keeping in mind that there could be multiple solutions based on periodic properties of cosine. --- Some famous mathematicians have made crucial contributions to understanding continuity, notably Augustin-Louis Cauchy, who formalized the concept of limits. His work paved the way for more comprehensive definitions in calculus, particularly the use of \( \epsilon \)-\( \delta \) proofs for continuity—essentially laying the mathematical groundwork for our modern understanding of continuous functions! Another interesting point is that continuous functions like this one play a pivotal role in real-world applications, like in physics and engineering. For instance, when modeling wave phenomena, both the sine and cosine functions are frequently involved. Understanding continuity helps in predicting how these waves behave as they travel through different media, ensuring accurate predictions in designing structures like bridges or buildings.