Find the largest region in the \( x y \)-plane in which the following function is continuous. \[ f(x, y)=\ln (x-9 y) \] Click for List \[ \text { Submit Assignment Quit \& Save } \]

To determine where the function \( f(x, y) = \ln(x - 9y) \) is continuous, we look for the conditions under which the argument of the logarithm is positive, since the natural logarithm is only defined for positive arguments. Thus, we need: \[ x - 9y > 0 \] This leads us to the inequality: \[ x > 9y \] The region in the \( xy \)-plane that satisfies this inequality is above the line \( x = 9y \). This line has a slope of \( \frac{1}{9} \) and passes through the origin, dividing the plane into two distinct regions. The area where \( f(x,y) \) is continuous is therefore the half-plane above the line defined by \( x = 9y \). The largest region where \( f \) is continuous is not open-ended, as for every point in this region, there is a corresponding point on the line where the function is undefined. You can visualize this as a slanted band extending infinitely in the positive \( x \) and \( y \) directions, but avoiding the boundary \( x = 9y \). To further clarify, this region includes all points where you can draw vectors from the origin that make an angle steeper than that of the line \( y = \frac{1}{9}x \). Therefore, any point above the line is where the function is happily defined!