Problem 50 - Calculus Early Transcendentals

Question

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$h(x, y)=\frac{\sqrt{x-y}}{4}$$

Step-by-Step Solution

Answer

The given function, \(h(x, y) = \frac{\sqrt{x-y}}{4}\), is continuous in the region where \(x \geq y\). That means it is continuous at any point \((x, y)\) in \(\mathbb{R}^{2}\) such that \(x \geq y\).

Step 1: Understand the function

Given the function: $$ h(x, y)=\frac{\sqrt{x-y}}{4} $$ Our goal is to find where this function is continuous.

Step 2: Analyze the denominator

The denominator is \(4\) which is never equal to \(0\). Therefore, we don't need to worry about it introducing any discontinuities.

Step 3: Analyze the square root

The square root can cause discontinuities if its argument is negative. Since we want to find points at which the square root is continuous, we should look for values of \(x\) and \(y\) for which: $$ x - y \geq 0 $$

Step 4: Solve the inequality

Solve the inequality to determine the region where the function is continuous: $$ x - y \geq 0 \Rightarrow x \geq y $$ The function is continuous in the region where \(x \geq y\). This translates to any \((x, y)\) in \(\mathbb{R}^{2}\) such that \(x \geq y\).