91Ó°ÊÓ

Continuity at a point is a fundamental concept in calculus and analysis, which indicates how a function behaves in close proximity to that point. For a function to be continuous at a point

• the function value must exist at that point,
• the limit of the function as it approaches that point must exist, and
• the function’s limit must equal its actual value at the point.

Employing the epsilon-delta definition of continuity, particularly highlights how we can quantify this behavior:If for every \( \varepsilon > 0, \) there exists a \( \delta > 0 \)such that whenever \( \sqrt{x^2 + y^2} < \delta, \)we have \( \left| f(x, y) - f(0,0) \right| < \varepsilon.\)In this exercise, we apply this concept to demonstrate the continuity of a bi-variable function at the origin (0,0). The task essentially involves showing that the function does not suddenly "jump" but instead, smoothly approaches a given value at the point as the input values approach the origin.

Handling expressions with two variables is often simplified by utilizing polar coordinates, especially when dealing with radial symmetry or problems involving a point (like the origin) in a two-dimensional plane. Polar coordinates express the same position as Cartesian coordinates but use a distance from the origin, \( r, \) and an angle, \( \theta. \) Converting from Cartesian to polar involves using:

• \( x = r\cos\theta \)
• \( y = r\sin\theta \)

Applying this to our function\( f(x, y) = \frac{xy^2}{x^2 + y^2}, \) it transforms into\( f(r\cos\theta, r\sin\theta) = \cos\theta \sin^2\theta. \)Thanks to the conversion, the function's relation to \( r, \) the distance from the origin, vanishes. This signifies that within some limits, the behavior of the function merely depends on the angle \( \theta. \) Converting to polar coordinates aids in visualizing how \( f(x, y) \) changes as the distance \( r \) tends to zero, centering the analysis on angular behavior rather than radial distance.

Analyzing how a function behaves near a particular point is crucial to understanding continuity. With the function expressed in polar coordinates, \( f(r\cos\theta, r\sin\theta) = \cos\theta \sin^2\theta, \)the real task is to investigate how this behaves as \( r \rightarrow 0.\)Since it no longer depends on \( r, \) we focus on the trigonometric portion, \( \cos\theta \sin^2\theta, \) to assess continuity.

• The trigonometric identities help us evaluate the function's maximum possible values: \( \sin\theta \) varies between -1 and 1,\
• and \( \cos\theta \) between -1 and 1.

Through this, we ensure that as \( r \rightarrow 0, \) the fluctuations in value are bounded, which determines how the radial limits affect the function’s proximity to a constant behavior around the origin. It is the interplay of these variables that guides our understanding of its behavior and helps set the stage for determining appropriate limits (i.e., \( \delta \)).

Finding the right balance of conditions within an epsilon-delta proof often leans heavily on inequality analysis -- a way to bound expressions to meet specified criteria. Given our function's form, \( \frac{xy^2}{x^2 + y^2} \leq \frac{1}{4} \)for \( \cos\theta \sin^2\theta \) values, we need to ensure it never exceeds the target \( \varepsilon = 0.04. \)

• This involves refining our bounds so \( |f(x, y) - 0| < \varepsilon \) holds true.
• One approach is conditioning \( y^2 < 4\epsilon x^2\) to maintain control over the output’s size.

By defining \( \delta \) based on this thought process, say letting \( \delta = \sqrt{4\varepsilon}, \)we achieve a radius within which the function’s deviation stays compact. This adjustment ensures the function stays below the \( \varepsilon \) threshold, distinctly evidencing continuity at the origin as required by the epsilon-delta criterion.