Problem 3 - Advanced Calculus

Question

Check that the function defined in \((2-2)\) is such that it is convergence preserving for all sequences of the form \(p_{n}=\left(a / n^{2}, b / n^{2}\right)\). 4 Let \(f\) be defined by \(f(x, y)=x^{2} y^{2} /\left(x^{2}+y^{2}\right)\), with \(f(0,0)=0\). By checking various sequences. test this for continuity at \((0,0)\). Can you tell whether or not it is continuous there?

Step-by-Step Solution

Answer

The function \(f(x, y)=x^{2} y^{2} /\left(x^{2}+y^{2}\right)\) is not continuous at (0,0).

Step 1: Examine the function at (0,0)

For the given function \(f(x, y)=x^{2} y^{2} /\left(x^{2}+y^{2}\right)\), it is clearly defined that \(f(0,0)=0\). Therefore, if the function is continuous at (0,0), any sequence that converges to (0,0) should also have \(f(x_n, y_n)\) converging to 0.

Step 2: Check sequences of the form \(p_n = \left(a / n^{2}, b / n^{2}\right)\)

For sequences of \(p_n = (a/n^2, b/n^2)\), where a and b are any real numbers, the corresponding function values are \(f(p_n) = a^2 b^2 / (a^2 + b^2)\). As n approaches infinity, \(p_n\) approaches (0,0) and \(f(p_n)\) should approach zero.

Step 3: Confirm with further sequences

Other sequences converging to (0,0) should also be checked. For example, a sequence of the form \(p_n = (1/n, 1/n)\) results in \(f(p_n) = 1/2\), not converging to zero. Therefore, it can be concluded that the function is not continuous at (0,0).

Key Concepts

Convergence of Sequences

Understanding the concept of convergence of sequences is fundamental in calculus. It involves sequences of numbers approaching a particular point as the number of terms grows large. For instance, consider the sequence given by the formula \(p_{n}=\left(a / n^{2}, b / n^{2}\right)\). As the term count (\