91Ó°ÊÓ

In real analysis, the concept of a limit is fundamental to understanding how functions behave as they approach certain points. Simply put, the limit describes the value that a function approaches as the input gets closer and closer to a specific point. For a function of two variables, such as \( f(x, y) = \frac{xy^2}{x^2 + y^2} \), you are investigating what happens when \((x, y)\) gets arbitrarily close to the origin \((0, 0)\).
Ultimately, you want \( f(x, y) \) to approach a target value, typically \( f(0, 0) \), which is zero in this case. By focusing on the distance \( \sqrt{x^2 + y^2} \), you're effectively examining how the function should behave as \((x, y)\) approaches \( (0, 0) \) from any direction. In our exercise, we determine a certain range, a small disk of radius \( \delta \), that ensures the function remains within a specific \( \epsilon \) range from zero.

The Epsilon-Delta definition formalizes what it means for a function to have a limit at a point. When we say "for every \( \epsilon > 0 \), there exists a \( \delta > 0 \)", we are building a bridge between how close \( f(x, y) \) gets to the limit value and how close \( (x, y) \) gets to the point of interest.

Here's how it works in practice:

• Pick a positive number \( \epsilon \) (e.g., 0.04 from our exercise) which represents a tiny window of tolerance around the target limit value of the function.
• You seek a \( \delta \) such that whenever \( \sqrt{x^2 + y^2} < \delta \), the value of the function \( |f(x, y) - f(0, 0)| < \epsilon \).

This approach means if \( (x, y) \) is within the distance \( \delta \) from \( (0, 0) \), then the function's value stay within \( \epsilon \) from zero. In our case, by choosing \( \delta = 0.02 \), the given epsilon condition is satisfied, ensuring the function behaves as expected.

Continuity is a key aspect of how smoothly a function behaves when its inputs change slightly. A continuous function doesn't leap or make sudden changes in value; rather, it gently transitions from one value to another.
In the context of this exercise, we seek continuity around the point \((0, 0)\), meaning we want the function \( f(x, y) = \frac{xy^2}{x^2 + y^2} \) to not behave erratically as \( (x, y) \) approaches the origin.

To achieve this:

• The function must be defined at the point \( (0, 0) \), which it is, with \( f(0, 0) = 0 \).
• The limit of \( f(x, y) \) as \( (x, y) \) approaches \( (0, 0) \) must equal \( f(0, 0) \).
• The epsilon-delta criteria need to be satisfied.

By proving there exists a \( \delta \) such that any point within the disk of radius \( \delta \) around the origin keeps the function \( f(x, y) \) close to zero, we confirm its continuity at the origin. This careful analysis ensures that the function acts predictably and without abrupt changes near \((0, 0)\).