Problem 43 - Calculus Early Transcendentals

Question

Graph and discuss the continuity of the function $$f(x, y)=\left\\{\begin{array}{ll}{\frac{\sin x y}{x y}} & {\text { if } x y \neq 0} \\ {1} & {\text { if } x y=0}\end{array}\right.$$

Step-by-Step Solution

Answer

The function \( f(x, y) \) is continuous everywhere since the limit matches the defined value at points where \( xy = 0 \).

Step 1: Understanding the Function Definition

The function \( f(x, y) \) is defined piecewise: - If \( x \, y eq 0 \), then \( f(x, y) = \frac{\sin(xy)}{xy}\).- If \( x \, y = 0 \), then \( f(x, y) = 1 \).The problem lies with the piecewise transition around the point where \( xy = 0 \), and we need to analyze its continuity there.

Step 2: Analyzing Continuity at Points Where \( xy \neq 0 \)

For points \( (x, y) \) where \( xy eq 0 \), the function \( f(x, y) = \frac{\sin(xy)}{xy} \) is a result of the limit definition of the derivative of \( \sin \) divided by its argument, hence continuous in its own domain.

Step 3: Investigating Continuity at Points Where \( xy = 0 \)

To check continuity at \( xy = 0 \), we need to see if the limit of \( f(x, y) \) as \( (x, y) \to (a, b) \) where \( ab = 0 \) equals to 1:\[ \lim_{(x,y)\to (a,b)} \frac{\sin(xy)}{xy} = 1 \].Given that \( \frac{\sin(t)}{t} \to 1 \) as \( t \to 0 \), the limit as \( (x, y) \to (a, b) \) where \( ab = 0 \) behaves similarly using the substitution \( t = xy \).

Step 4: Verifying Limit Existence in Directional Approaches

Let's approach the origin along different paths:1. **Along the x-axis**: Set \( y = 0 \), \( f(x, 0) = 1 \).2. **Along the y-axis**: Set \( x = 0 \), \( f(0, y) = 1 \).3. **Along \( y = mx \)*: \( f(x, mx) \) results in \( \frac{\sin(m x^2)}{m x^2} \to 1 \) as \( x \to 0 \) since it matches the derivative limit of sine.Different paths all lead to the limit of 1, verifying continuity there.

Step 5: Conclusion on Continuity

The function \( f(x, y) \) is continuous everywhere, because at every point where \( xy = 0 \), the value \( f(x, y) = 1 \) coincides with the limit from all approaching paths as seen in the previous step.

Key Concepts

Piecewise Functions

Piecewise functions are a type of function defined by different expressions depending on the inputs. In our context, the function \( f(x, y) \) uses different formulas based on the product of \( x \) and \( y \).
If \( x y eq 0 \), the function is given by:

• \( f(x, y) = \frac{\sin(xy)}{xy} \)

If \( x y = 0 \), the function becomes a constant:

• \( f(x, y) = 1 \)

Understanding how these pieces come together is crucial for analyzing the behavior of the function near specific points, particularly where the transition between the two cases happens.

Limit Definition

The concept of limits is vital in understanding continuity in multivariable functions. The limit definition helps us find the value that the function approaches as the input points get arbitrarily close to a specific point.
To determine the continuity of \( f(x, y) \), we need to evaluate the limit as \( (x, y) \) approaches points where \( xy = 0 \). This checks whether the function's behavior aligns with its definition at those points.
With \( \lim_{(x,y)\to (a,b)} \frac{\sin(xy)}{xy} = 1 \), we can confirm the function's continuity by ensuring the limit matches the defined value when \( xy = 0 \). This is done by substituting \( t = xy \) and leveraging the fact that \( \frac{\sin(t)}{t} \to 1 \) as \( t \to 0 \).

Derivative of Sine

The derivative of sine plays a significant role when analyzing the continuity of our function along paths where \( xy eq 0 \).
The expression \( \frac{\sin(xy)}{xy} \) resembles the derivative or slope of \( \sin(t) \) at zero. It is a classic limit example: as \( t \to 0 \), \( \frac{\sin(t)}{t} \to 1 \).
This property confirms the continuity of the function \( f(x, y) \) at all points in its domain, except at the boundary where \( xy = 0 \), thus connecting to the limits we evaluate for continuity analysis.

Directional Limits

Directional limits are used to analyze the behavior of functions as they approach a point from different directions.
For the function \( f(x, y) \), determining continuity at \( xy = 0 \) relies on checking different paths approaching this boundary:

• **x-axis**: \( f(x, 0) = 1 \), since \( y = 0 \)
• **y-axis**: \( f(0, y) = 1 \), since \( x = 0 \)
• Using the path **y = mx**: \( f(x, mx) = \frac{\sin(mx^2)}{mx^2} \to 1 \) as \( x \to 0 \)

On all these paths, the limit leads to the same value 1, hence ensuring that \( f(x, y) \) remains continuous as it transitions across \( xy = 0 \).