Problem 57 - Advanced Calculus

Question

Let \(f: R^{2} \rightarrow R\) be given by $$ f(x, y)=x^{2}+y^{2} $$ Show that \(\mathrm{f}\) is continuous at \((0,0)\).

Step-by-Step Solution

Answer

Using the limit definition of continuity, we demonstrated that the value of the function \(f(x, y) = x^2 + y^2\) at \((0,0)\) is equal to the limit of the function as \((x, y)\) approaches \((0,0)\). By converting to polar coordinates and finding the limit, we showed that \(f(0,0) = \lim_{(x, y) \to (0,0)} f(x, y) = 0\), thus proving \(f(x, y)\) is continuous at \((0,0)\).

Step 1: Find the value of the function at the point \((0,0)\)

To find the value of \(f(x, y)\) at the point \((0,0)\), substitute \(x = 0\) and \(y = 0\) into the expression for \(f(x, y)\): $$ f(0, 0) = (0)^2 + (0)^2 = 0 $$

Step 2: Determine the limit of the function as \((x, y)\) approaches \((0,0)\)

We must show that the limit of \(f(x, y)\) as \((x, y)\) approaches \((0,0)\) exists and equals the value of the function at the point \((0,0)\). Using polar coordinates, let \(r = \sqrt{x^2 + y^2}\) and \(\theta\) be any angle. Then, the limit can be expressed as: $$ \lim_{(x, y) \to (0,0)} f(x, y) = \lim_{r \to 0} r^2 $$

Step 3: Evaluating the limit

To find the limit, we simply substitute \(r = 0\): $$ \lim_{r \to 0} r^2 = 0^2 = 0 $$

Step 4: Comparing the function value and the limit

Now that we have found both the value of the function at the point \((0,0)\), which is \(f(0,0)=0\), and the limit as \((x, y)\) approaches \((0,0)\), which is \(\lim_{(x, y) \to (0,0)} f(x, y) = 0\), we can conclude that: $$ f(0,0) = \lim_{(x, y) \to (0,0)} f(x, y) $$ Since the value of \(f(x, y)\) at \((0,0)\) is equal to the limit of \(f(x, y)\) as \((x, y)\) approaches \((0,0)\), the function is continuous at \((0,0)\).

Key Concepts

Limit of a Function

Understanding the limit of a function is crucial to determine continuity. Limits help us predict the behavior of a function as the input values approach a specific point.
When dealing with functions of multiple variables, such as two variables in this exercise, we need to analyze how the function behaves as both variables approach some given values simultaneously. For a function \(f(x, y)\), the limit as \((x, y)\) approaches a point \((a, b)\) is a value \(L\) that approaches the expected behavior of the function near \((a, b)\). To prove continuity at \((0,0)\), it's necessary to demonstrate that:

• The limit \(\lim_{(x, y) \to (0,0)} f(x, y)\) exists.
• The limit is equivalent to the function value at the point: \(f(0,0)\).

Only if both these conditions hold, can the function be considered continuous at the specified point.

Polar Coordinates

Polar coordinates are an alternative to the Cartesian coordinate system. This method simplifies calculations when dealing with multivariable functions, especially in problems involving symmetry like ours.
In polar coordinates, any point \((x, y)\) is represented using \(r\) (the radial distance from origin) and \(\theta\) (the angle from the positive x-axis). The transformation is given by:

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

In our exercise, replacing \(x\) and \(y\) using these relations simplifies the calculation of the limit:

• \(f(x, y) = x^2 + y^2\) becomes \(r^2(\cos^2(\theta) + \sin^2(\theta))\).
• Since \(\cos^2(\theta) + \sin^2(\theta) = 1\), the expression simplifies to \(r^2\).

This transformation verifies that as \(r \to 0\), the function \(f(x, y) = r^2\) approaches zero, aligning with finding the limit and proving continuity at \((0,0)\).

Multivariable Functions

Multivariable functions involve multiple inputs, often requiring a different approach than single-variable functions.
The function \(f(x, y) = x^2 + y^2\) in our exercise takes two inputs, \(x\) and \(y\), and therefore, involves evaluating behavior in two-dimensional space, not a single axis. Determining continuity for such functions demands we consider all paths leading to a given point.
By using polar coordinates, we simplified this examination, but typically, ensuring continuity means:

• Checking the limit exists and is the same along multiple paths (like lines or curves) leading towards the point.
• Verifying the obtained limit matches the function's value at that exact point.

Only by ensuring these conditions can we confidently say a multivariable function is continuous at any given point.