Question

Prove: If \(f_{x}\) and \(f_{y}\) are bounded in a neighborhood of \(\left(x_{0}, y_{0}\right),\) then \(f\) is continuous at \(\left(x_{0}, y_{0}\right)\).

Short answer

Explain how to prove that if both the partial derivatives of a function f are bounded in a neighborhood around a point, then the function is continuous at that point. To prove that if both the partial derivatives of a function f are bounded in a neighborhood around a point, then the function is continuous at that point, follow these steps: 1. Define the linear approximation of f at the point in question, which is given by the formula: \( L(x, y) = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \). 2. Use the Mean Value Theorem for functions of several variables to relate f and its linear approximation, resulting in: \( f(x, y) - L(x, y)= K (x-x_0) + M (y-y_0)\), where K and M are constants. Note that since f_x and f_y are bounded, we know that |K|, |M| are also bounded by constants. 3. Show that f is continuous at the point based on the relation derived in step 2 by computing the limit of \(|f(x, y) - f(x_0, y_0)|\) as \((x, y) \rightarrow (x_0, y_0)\). Use the fact that both f_x and f_y are bounded, which allows us to show that the limit goes to 0 as \((x, y) \rightarrow (x_0, y_0)\). This proves that f is continuous at the point.

Step-by-step solution

Define the linear approximation of f

The linear approximation of the function f at a point \((x_0,y_0)\) is given by the formula: $$ L(x, y) = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) $$

Use Mean Value Theorem to relate f with its linear approximation

By applying the Mean Value Theorem for functions of several variables, we can relate f and its linear approximation as follows: $$f(x, y) - L(x, y)= K (x-x_0) + M (y-y_0)$$ where \(K\) and \(M\) are constants and $$K = f_x(\xi, \eta)(x - x_0), \quad M = f_y(\xi, \eta)(y - y_0)$$ for some point \((\xi, \eta)\) between \((x_0, y_0)\) and \((x, y)\). Note that since \(f_x\) and \(f_y\) are bounded, we know that there exists a constant \(C\) such that \(|K|, |M| \le C|f_x(x_0, y_0)|, C|f_y(x_0, y_0)|\).

Show that f is continuous at \((x_0, y_0)\) based on the relation we derived

Now we shall compute the limit of \(|f(x, y) - f(x_0, y_0)|\) as \((x, y) \rightarrow (x_0, y_0)\): $$ |\lim_{(x, y) \to (x_0, y_0)} (f(x, y) - f(x_0, y_0))| = \lim_{(x, y) \to (x_0, y_0)} |f(x, y) - L(x, y)| $$ From step 2, we know that \(|f(x,y)-L(x,y)|=|K(x-x_0)+M(y-y_0)| \le |K(x-x_0)|+|M(y-y_0)|\), and we also know that \(|K|, |M| \le C|f_x(x_0, y_0)|, C|f_y(x_0, y_0)|\). So, we get that the limit is bounded by: $$ \lim_{(x, y) \to (x_0, y_0)} (C|f_x(x_0, y_0)|(x - x_0) + C|f_y(x_0, y_0)|(y - y_0))$$ which goes to 0 as \((x, y) \rightarrow (x_0, y_0)\) since both \(f_x\) and \(f_y\) are bounded. Therefore, we have proven that if \(f_x\) and \(f_y\) are bounded in a neighborhood of \((x_0, y_0)\), then f is continuous at \((x_0, y_0)\).

Key concepts

Bounded Partial Derivatives

Understanding the concept of bounded partial derivatives is fundamental in the study of real analysis, especially when dealing with the continuity of multivariable functions. When we say the partial derivatives of a function are bounded, we mean there exists an upper limit, or bound, for their values within a certain neighborhood around a point.

For a function with two variables, such as our function f, this means that the rates of change of f with respect to x and y, which are denoted as \( f_x \) and \( f_y \), respectively, do not exceed certain fixed values no matter how close to the point \( (x_0, y_0) \) we get. This restriction on growth rate is a key property that contributes to the predictable behavior of a function and paves the way to proving continuity.

Mean Value Theorem

The Mean Value Theorem for multivariable functions is an extension of the classic theorem from single-variable calculus. This principle provides a link between the derivatives of a function and the function's overall behavior over an interval or, in our case, over a region in the xy-plane.

The theorem implies that for two points in the domain of a function, there's at least one point between them where the partial derivatives \( f_x \) and \( f_y \) take on certain values, denoted as K and M in the solution. These values explain the difference between the function and its linear approximation, L, at any point within the neighborhood. By knowing that the function's derivatives are bounded, and applying this theorem, we can infer a lot about the function's behavior, including whether or not it is continuous at a particular point.

Multivariable Functions

Functions of several variables, like the one in our exercise, can seem daunting at first. These are functions that depend on two or more variables, and their continuity cannot be determined by looking at the behavior in just one direction.

To study these functions, we examine changes in all directions from a particular point in their domain. In real analysis, understanding the behavior of these functions near a point involves considering all the paths one could take to approach that point. The concept of bounded partial derivatives ensures that no matter which path is taken, the change in the function's value is not erratic, supporting a key outcome, like continuity, at that point.

Linear Approximation

A linear approximation is a strategy to approximate the value of a function near a specific point using a linear function, or plane in the case of two variables, that closely matches the function's value and slope at that point.

This approximation, denoted as L(x, y), is often the first step to describe the behavior of a function around a point because it simplifies complex functions into more manageable linear forms. By comparing a function to its linear approximation, as seen in the step-by-step solution, we can measure how a function deviates from this linearity. This measure helps us evaluate the continuity of the function because if the difference between the actual function and the linear approximation approaches zero as we get closer to the point, continuity at that point is confirmed.