Question

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$p(x, y)=e^{x-y}$$

Short answer

Answer: Yes, the function \(p(x, y) = e^{x-y}\) is continuous at all points in \(\mathbb{R}^2\).

Step-by-step solution

Find the Function's Limit

To find the limit of \(p(x, y)\), as \((x, y) \rightarrow (a, b)\), take the limit by substituting the expression for the function: $$\lim_{(x, y) \rightarrow (a, b)} p(x, y) = \lim_{(x, y) \rightarrow (a, b)} e^{x-y}.$$

Manipulate the Limit

Now, let's rewrite the limit in terms of the exponential function. Recall that the exponential function is continuous for all real numbers: $$\lim_{(x, y) \rightarrow (a, b)} e^{x-y} = e^{\lim_{(x, y) \rightarrow (a, b)} (x-y)}.$$

Evaluate the Inner Limit

Evaluate the limit of the difference \((x-y)\) as \((x, y) \rightarrow (a, b)\): $$\lim_{(x, y) \rightarrow (a, b)} (x-y) = \lim_{x \rightarrow a} x - \lim_{y \rightarrow b} y = a - b.$$

Evaluate the Exponential Limit

Now, substitute the result of Step 3 into the limit from Step 2: $$e^{\lim_{(x, y) \rightarrow (a, b)} (x-y)} = e^{a-b}.$$

Check Continuity

A function is continuous at \((a, b)\) if the limit of the function as \((x, y) \rightarrow (a, b)\) is equal to the function's value at \((a, b)\). So, let's check if the function's value at \((a, b)\) is equal to the limit we found: $$p(a, b) = e^{a-b}.$$ Since the limit of the function as \((x, y) \rightarrow (a, b)\) is equal to the function's value at \((a, b)\), we can conclude that the function is continuous at all points in \(\mathbb{R}^2\).

Key concepts

Limits in Calculus

In order to understand the behavior of functions at specific points, especially when they involve more than one variable, we turn to limits. The concept of a limit in calculus is pivotal for determining continuity and analyzing function behavior as a point approaches a particular value.

For multivariable functions like the exemplary function given by \( p(x, y) = e^{x-y} \), the limit as \( (x, y) \) approaches a point \( (a, b) \) is essential to define. It is the value that \( p(x, y) \) gets closer to as \( x \) approaches \( a \) and \( y \) approaches \( b \).

Technically, to find the limit of a multivariable function, you have to ensure that the function approaches the same value irrespective of the path taken to get to the point \( (a, b) \). In the given example, the function \( p(x, y) \) simplifies to an exponential function which is known to be continuous for all real number inputs; hence, simplifying our work in verifying the limit.

With single-variable functions, determining limits is often a straightforward process, but with multiple variables, care must be taken to consider all possible approaches to the point in question. This is particularly relevant for functions that are defined piecewise or have different behavior along different paths.

Exponential Functions

Exponential functions are a class of mathematical functions characterized by an equation of the form \( f(x) = a^x \), where \( a \) is a constant called the base, and \( x \) is the exponent. These functions are fundamental in various branches of mathematics, including calculus, due to their unique properties.

One of the remarkable traits of exponential functions is their continuity. That is, they do not have any breaks, jumps, or holes in their graphs and are defined for all real numbers, which is why they are used extensively in modeling continuous growth or decay processes like population growth, radioactive decay, and compound interest.

In our example \( p(x, y) = e^{x-y} \), the base is the natural exponential \( e \), which is an irrational constant approximately equal to 2.71828. The function is continuous everywhere in its domain, which includes all real numbers, as there are no values of \( x \) or \( y \) for which it is undefined. Its continuity is a crucial aspect when examining the limits and behavior of the function at various points across the real plane \( \( \mathbb{R}^{2} \) \).

Continuity in Mathematics

Continuity of a function is a fundamental concept in mathematics that ties in directly with the topics of limits and exponential functions. A function is said to be continuous at a point if you can draw the function at this point without lifting your pen from the paper. Formally, a function \( f \) is continuous at a point \( c \) if the limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \).

For a multivariable function like \( p(x, y) = e^{x-y} \), the definition is analogous. The function is continuous at a point \( (a, b) \) if the limit as \( (x, y) \) approaches \( (a, b) \) is equal to the function's value at \( (a, b) \), which, as we found in the step-by-step solution, is indeed the case for our example.

It's essential to note that continuity is not just about the function being well-behaved at a single point but is about its behavior in a neighborhood of that point. In other words, for a function to be continuous at a point, it must be defined at that point, and there should be no abrupt changes in its behavior in the immediate vicinity of the point. The elegance of the exponential function in the exercise confirms its continuity across the entire real plane, embodying a smooth and predictable behavior that makes it a valuable tool for mathematical modeling.