Question

Suppose that \(f\) is a real-valued function defined in an open set \(E \subset R^{n}\), and that the partial derivatives \(D_{1} f, \ldots, D_{n} f\) are bounded in \(E\). Prove that \(f\) is continuous in \(\bar{E}\) Hint: Proceed as in the proof of Theorem 9.21.

Short answer

Based on the given step by step solution, construct a short answer question: Question: Prove that a real-valued function \(f\) defined on an open set \(E \subset R^{n}\) with bounded partial derivatives is continuous in \(\bar{E}\), using a similar proof to Theorem 9.21. Answer: To prove the continuity of \(f\), follow these steps: 1. Write the difference \(f(y) - f(x)\) in terms of the partial derivatives using Taylor's theorem for several variables. 2. Apply the mean value theorem to find an intermediate point \(\xi_i\) for each partial derivative \(D_i f(\xi_i)\) between \(x_i\) and \(y_i\). 3. Use the triangle inequality to compare the absolute value of the sum of the products of the bounded partial derivatives and the differences between the corresponding components of \(x\) and \(y\). 4. Show that the sum of the products goes to zero as \(y\) approaches \(x\), proving the continuity of \(f\) in \(\bar{E}\).

Step-by-step solution

Express the difference in terms of partial derivatives

First, we write the difference \(f(y) - f(x)\) using Taylor's theorem for several variables: $$f(y) - f(x) = \sum_{i=1}^{n} D_i f(\xi_i)(y_i - x_i) \ ,$$ where \(\xi_i\) is a point on the segment joining \(x\) and \(y\).

Apply the mean value theorem

Using the mean value theorem, we can find an intermediate point \(\xi_i\) for each partial derivative \(D_i f(\xi_i)\) between \(x_i\) and \(y_i\). Now, we only need to consider the difference between these two points as the real numbers \((y_i - x_i)\): $$\sum_{i=1}^{n} D_i f(\xi_i)(y_i - x_i)$$

Apply the triangle inequality

To proceed, we will use the triangle inequality: $$\left|\sum_{i=1}^{n} D_i f(\xi_i)(y_i - x_i) \right| \leq \sum_{i=1}^{n} \left|D_i f(\xi_i)(y_i - x_i)\right|$$

Use the fact that partial derivatives are bounded

Now, we apply the fact that the partial derivatives are bounded and continuous in \(\bar{E}\). $$\sum_{i=1}^{n} \left|D_i f(\xi_i)(y_i - x_i)\right| \leq \sum_{i=1}^{n} M_i |y_i - x_i| \ ,$$ where \(M_i\) are the upper bounds of \(|D_i f(\xi_i)|\) on \(\bar{E}\). Finally, we let \(y\) approach \(x\) and observe that the sum \(\sum_{i=1}^{n} M_i |y_i - x_i|\) approaches zero: $$\lim_{y \to x} \sum_{i=1}^{n} M_i |y_i - x_i| = 0$$ Hence, by the squeeze theorem: $$\lim_{y \to x} |f(y) - f(x)| = 0$$ This concludes the proof that \(f\) is continuous in \(\bar{E}\).

Key concepts

Partial Derivatives

In mathematical analysis, partial derivatives play a central role when it comes to functions of several variables. The partial derivative of a function is its derivative with respect to one variable, while holding the other variables constant. It measures the rate at which the function changes as only that particular variable changes. In the given exercise, the partial derivatives of the function, noted as \(D_{1} f, \.\.\.\,\, D_{n} f\), are fundamental in establishing the continuity of the function over the set \(E\). Since they are bounded, it implies that for each direction there is a limit to how much the function can change, supporting the idea that the function behaves consistently as points approach one another within that set.

When proving a function's continuity using partial derivatives, one effective approach is to show that small changes in the input points result in small changes in the function's output. This concept is visually represented by the slopes of the function not reaching extreme values, which is why bounded partial derivatives often lead to continuous functions. It's important to note, the existence of partial derivatives alone does not automatically imply continuity, but boundedness helps to strengthen the case.

Mean Value Theorem

The mean value theorem is a fundamental result in calculus, asserting that if a real-valued function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point where the function's derivative is equal to the average rate of change over that interval. In the context of the exercise, the mean value theorem is applicable in a higher-dimensional setting through its analogous form for functions of several variables. It tells us that there exists a point \(\xi_i\) between \(x_i\) and \(y_i\) for each partial derivative \(D_i f\).

The theorem assists in relating the change in the function's value to the behavior of its derivatives and simplifies the expression of the difference \(f(y) - f(x)\). Essentially, the mean value theorem is the bridge between local behavior, described by derivatives, and the global change in function values. This exercise demonstrates how the theorem is essential in proving the continuity of a function by linking the boundedness of partial derivatives to the controlled variation of the function.

Triangle Inequality

The triangle inequality is a principle from geometry that also serves as an indispensable tool in analysis. It states that for any real numbers or vectors, the sum of their lengths is always greater than or equal to the length of their sum. In a more formal expression for real numbers, \(\left|a + b\right| \leq \left|a\right| + \left|b\right|\). When dealing with the sum of function changes encoded by the partial derivatives in our exercise, the triangle inequality is utilized in Step 3 to transition from the absolute value of the sum to the sum of absolute values. This step is crucial as it transforms the problem into a sum of simpler terms which can be easily bounded.

By breaking down the complex behavior of \(f\) across multiple variables into simpler one-dimensional scenarios, the triangle inequality lets us build an argument for continuity piece by piece. It ensures that even as we accumulate the changes along each dimension, the overall change in the function's value won't spiral out of control, aligning with the idea of continuity where small changes in input yield small changes in output.

Taylor's Theorem

Taylor's theorem gives us an approximation of a function around a specific point using polynomials derived from the function's derivatives. For functions of several variables, Taylor's theorem can be used to express the function value at a nearby point \(y\) in terms of its value at \(x\) and a linear combination of the changes in the variables weighted by the partial derivatives at intermediate points. The theorem essentially allows us to approximate the behavior of \(f\) near \(x\) with the sum \(\sum_{i=1}^{n} D_i f(\xi_i)(y_i - x_i)\) as seen in Step 1 of the solution.

By providing a way to linearize the function's local change, Taylor's theorem sets the stage for bounding this change and hence proving continuity. The more precise and controlled these local linear approximations are, as ensured by the boundedness of the partial derivatives, the more we can guarantee the smallness of the change in value of the function, culminating in the proof of continuity through the squeeze theorem.