Problem 72 - Advanced Calculus

Question

Distinguish between removable and essential discontinuities. Decide, for the following functions whether the points of discontinuity are a) \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{xy} / \mathrm{x}^{2}+\mathrm{y}^{2} \quad(\mathrm{x}, \mathrm{y}) \in \mathrm{R}^{2}-\\{(0,0)\\}\) b) \(\mathrm{f}(\mathrm{x}, \mathrm{y})=\mathrm{x}^{2} \mathrm{y}^{2} / \mathrm{x}^{2}+\mathrm{y} \quad(\mathrm{x}, \mathrm{y}) \in \mathrm{R}^{2}-\\{(0,0)\\}\) c) \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{\mathrm{x}}\) \(x \in R, x>0\) d) \(g(x) \quad=x\) \(\mathrm{x} \in \mathrm{R}, \mathrm{x}>0\) and \(=2 \quad x \in R, x=0\) e) \(h(x)=\sin (1 / x) \quad x \in R, x<0\)

Step-by-Step Solution

Answer

In summary: a) Removable discontinuity at (0,0). b) Removable discontinuity at (0,0). c) No discontinuities. d) Essential discontinuity at \(x=0\). e) Essential discontinuity at \(x=0\).

Step 1: Analyze the function

For function a), we have: \(f(x, y) = \frac{xy}{x^2 + y^2} \quad (x, y) \in R^2 - \{(0, 0)\}\) The only point of discontinuity for this function is (0, 0) since it's excluded from the domain.

Step 2: Check if the discontinuity is removable

As \(x\) and \(y\) both approach 0, the function will become zero. Therefore, we have a removable discontinuity at (0, 0). 2) Function b)

Step 1: Analyze the function

For function b), we have: \(f(x, y) = \frac{x^2 y^2}{x^2 + y} \quad (x, y) \in R^2 - \{(0, 0)\}\)

Step 2: Check if the discontinuity is removable

As both \(x\) and \(y\) approach 0, the function becomes zero. Therefore, we also have a removable discontinuity at (0, 0). 3) Function c)

Step 1: Analyze the function

For function c), we have: \(f(x) = x^x \quad x \in R, x > 0\)

Step 2: Check for discontinuities

This function is continuous for all positive real numbers, so there are no discontinuities. 4) Function d)

Step 1: Analyze the function

For function d), we have: \(g(x) = x \quad x \in R, x > 0\) \(g(x) = 2 \quad x \in R, x = 0\)

Step 2: Check if the discontinuity is removable

This function has a jump discontinuity at \(x = 0\), which is essential since removing the point \(x = 0\) would not make it continuous. 5) Function e)

Step 1: Analyze the function

For function e), we have: \(h(x) = \sin(\frac{1}{x}) \quad x \in R, x < 0\)

Step 2: Check for discontinuities

This function is oscillating near \(x = 0\) and does not have a well-defined limit. As such, we have an essential discontinuity at \(x = 0\).

Key Concepts

Removable Discontinuities

Removable discontinuities occur when a function has a 'hole' in its graph that can be filled in by redefining the function at a single point. Imagine a simple path with a gap. If you can fill in that gap perfectly without affecting the rest of the path, that's a removable discontinuity.
For instance, both functions a) and b) from the original exercise have removable discontinuities at (0, 0). Here, the limit of the function's output as the point of discontinuity is approached matches the values around it. Therefore, by reassigning a specific value at that point, we can likely make the function continuous across the entire plane.

• Removable discontinuities imply that the limit exists as we approach the discontinuity.
• We can redefine the function at a point to remove the discontinuity.
• Analyzing limits at points of discontinuity helps ascertain their removability.

Essential Discontinuities

Essential discontinuities cannot be removed by simply redefining the function at one point. These are typically characterized by either jumps, oscillations, or vertical asymptotes in the graph. When approaching these points, the behavior of the function becomes unpredictable.
For example, function d) in the original exercise has a jump discontinuity at x = 0, illustrating an essential discontinuity. Similarly, function e) is highly oscillatory near x = 0, and its erratic behavior signals an essential discontinuity.

• Essential discontinuities mean the limit does not exist as you approach the point from different directions.
• There is no simple value that can be assigned to make the function continuous.
• Predictability is lost, making these discontinuities more complex.

Multivariable Functions

Multivariable functions give outputs based on two or more input variables, like x and y in this context. Visualizing these functions often involves imagining the surface that extends through three-dimensional space. Understanding discontinuities becomes more complex because the input approaches the discontinuity from multiple paths.
In the given exercise, functions a) and b) are examples of multivariable functions that have discontinuities at the origin (0,0). Since we can approach the point from numerous directions in the plane, identifying the nature of discontinuities in multivariable functions requires a thorough examination of different paths.

• Discontinuities in multivariable functions can be harder to identify.
• Limits are evaluated as the point is approached from various paths.
• Graphical representation can assist in identifying discontinuity types.

Discontinuity Analysis

Analyzing discontinuities involves determining whether they are removable or essential and understanding the behavior of the function near these points. This process requires a deep dive into the limits and the existence of these limits as we approach the point of discontinuity from various directions.
In discontinuity analysis, key steps include:

• Identifying the point where discontinuity occurs.
• Assessing the limit from different paths or directions.
• Determining the type of discontinuity by the behavior of the function: removable if a limit exists and matches the surrounding values; essential if no such limit exists.
• Considering the possibility of redefining the function to heal removable discontinuities.

Effective discontinuity analysis ensures a better grasp on the continuity of complex functions and aids in understanding their graphs fully.