Problem 9 - Calculus Early Transcendentals

Question

At what points of \(\mathbb{R}^{2}\) is a rational function of two variables continuous?

Step-by-Step Solution

Answer

Answer: A rational function of two variables is continuous at every point $(x, y) \in \mathbb{R}^2$ such that the denominator, $q(x, y) \neq 0$.

Step 1: Define a Rational Function

A rational function of two variables can be written in the form: $$f(x,y) = \frac{p(x,y)}{q(x,y)}$$ where \(p(x,y)\) and \(q(x,y)\) are both polynomial functions in the variables \(x\) and \(y\).

Step 2: Determine When the Denominator is Not Zero

A rational function is undefined when the denominator, \(q(x,y)\), is equal to zero. So, we need to find where \(q(x,y) \neq 0\). In general, this will require finding the roots of the polynomial equation \(q(x,y)=0\). However, we cannot provide an explicit solution for the roots of the given polynomial, as we do not have details on the specific polynomial. In general, finding the roots of a two-variable polynomial can be a complex task, and finding a general form for roots may not be possible.

Step 3: Rational Function Continuity

A rational function is continuous at any point where it is defined and has a limit. Since it is undefined when \(q(x, y) = 0\), we can say that the function is continuous everywhere in \(\mathbb{R}^{2}\) except where the denominator is equal to zero. So, the rational function \(f(x,y)\) is continuous at every point \((x, y) \in \mathbb{R}^2\) such that \(q(x, y) \neq 0\).