Problem 2 - Advanced Calculus

Question

Prove that \(Q(x, y)=x / y\) is continuous everywhere except on the line \(y=0\).

Step-by-Step Solution

Answer

Yes, the function \(Q(x, y)=x / y\) is continuous at all points except along the line \(y=0\) where the function is undefined.

Step 1: Apply the Definition of Continuity

Firstly, to prove that the function \(Q(x, y)=x / y\) is continuous, we must check that \(\lim_{{(x,y)} \to {(a,b)}} Q(x, y)=Q(a, b)\). That is, the value of the function at the point should be same as the limit of the function at that point.

Step 2: Evaluating the Limit

Now, let us compute the limit: \(\lim_{{(x,y)} \to {(a,b)}} Q(x, y)=\lim_{{(x,y)} \to {(a,b)}} (x / y)\). If \((a,b)\) is not on the line \(y=0\), we get \(\lim_{{(x,y)} \to {(a,b)}} (x / y) = a/b\) as \(y\) is not equal to zero under the limit.

Step 3: Checking the Equality

Now, we have to check whether \(\lim_{{(x,y)} \to {(a,b)}} Q(x, y)=Q(a, b)\). The right hand side is \(Q(a, b)=a / b\). Hence, \(\lim_{{(x,y)} \to {(a,b)}} Q(x, y)=Q(a, b)\) if \(y \neq 0\). Therefore, the function is continuous everywhere except on the line \(y=0\).

Key Concepts

Continuity

Continuity is a fundamental concept in calculus that describes how smooth and unbroken a function is. For a function to be continuous at a point, three conditions must meet:

• The function is defined at the point.
• The limit of the function as it approaches the point exists.
• The limit of the function is equal to the function's value at the point.

For the function \(Q(x, y) = \frac{x}{y}\), we need to ensure these conditions are satisfied for all points except on the line \(y=0\). In this case, if \(y\) is zero, \(Q(x, y)\) is not defined, disrupting continuity.
To check continuity, one must confirm that as \((x, y)\) gets infinitely close to a particular point \((a, b)\), the function \(Q(x, y)\) smoothly approaches the same value as \(Q(a, b)\). This alignment of the function's value with its limit is a hallmark of continuity, absent only when \(y=0\) for this function.

Limits

Limits are essential in understanding the behavior of functions as inputs get close to a particular value or "approach" a point. For the function \(Q(x, y) = \frac{x}{y}\), we evaluate the limit to verify its continuity.
When we express that \(\lim_{{(x,y)} \to {(a,b)}} Q(x, y)\) equals \(Q(a, b)\), we are determining if the function's output approaches a specific value as \((x, y)\) nears \((a, b)\). The process demands \((a, b)\) not be on the line \(y=0\), because division by zero is undefined, which means the limit wouldn’t exist at that line.
Getting the limit involves kind of "sneaking up" on the point \((a, b)\) from all directions in the plane. If different approaches to the point yield the same limit, then we can say confidently that the limit exists and contributes to the function's continuity at \((a, b)\) except where \(y=0\).

Functions of Two Variables

A function of two variables is a function that takes two inputs, often represented as \(x\) and \(y\), and gives a single output. The function \(Q(x, y) = \frac{x}{y}\) is an example of such a function. These types of functions can be visualized on a three-dimensional graph with both \(x\) and \(y\) as axes and the output along the third dimension.
Functions of two variables are more complex than single-variable functions due to the intricacies involved in their behavior across a plane. Continuity and limits for these functions involve checking how they behave not just as they approach a point along a line, but any direction across the plane.
Moreover, discontinuities can stem from restrictions like division by zero, as observed in this example. When \(Q(x, y)\) pivots around the line where \(y=0\), this results in breaks in the surface of the graph, highlighting where the function ceases to be defined.