Question

Suppose that \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) and \(h: \mathbb{R}^{n} \rightarrow \mathbb{R}\) have the same domain and are continuous at \(\mathbf{X}_{0}\). Show that the product \(h \mathbf{F}=\left(h f_{1}, h f_{2}, \ldots, h f_{m}\right)\) is continuous at \(\mathbf{X}_{\mathbf{0}}\).

Short answer

Question: Prove that the product \(h \mathbf{F}=(h f_1, h f_2, \ldots, h f_m)\) is continuous at \(\mathbf{X}_{\mathbf{0}}\) if functions \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) and \(h: \mathbb{R}^{n} \rightarrow \mathbb{R}\) are continuous at \(\mathbf{X}_{\mathbf{0}}\). Answer: The product \(G(\mathbf{X}) = h(\mathbf{X}) \mathbf{F}(\mathbf{X})\) is continuous at \(\mathbf{X}_{\mathbf{0}}\) as the product of continuous functions, each component of \(G(\mathbf{X})\), is also continuous at \(\mathbf{X}_{\mathbf{0}}\). Hence, the product \(h \mathbf{F}=(h f_1, h f_2, \ldots, h f_m)\) is continuous at \(\mathbf{X}_{\mathbf{0}}\).

Step-by-step solution

Define the product function

Define the product function \(G(\mathbf{X}): \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\) as \(G(\mathbf{X}) = h(\mathbf{X}) \mathbf{F}(\mathbf{X})\). In component form, we have \(G(\mathbf{X}) = (h(\mathbf{X})f_1(\mathbf{X}), h(\mathbf{X})f_2(\mathbf{X}), \ldots, h(\mathbf{X})f_m(\mathbf{X}))\).

Prove that the product function is continuous

To prove that \(G(\mathbf{X})\) is continuous, we will show that each component of the product function is continuous. Since the function \(h\) is continuous at \(\mathbf{X}_{0}\) and each component function \(f_i\) of \(\mathbf{F}\) is continuous at \(\mathbf{X}_{0}\), the product \(h(\mathbf{X})f_i(\mathbf{X})\) is continuous at \(\mathbf{X}_{0}\) for \(i = 1, \ldots, m\). This is due to the property that the product of continuous functions is also continuous.

Conclude that the product function is continuous at \(\mathbf{X}_{0}\)

Since each component function of the product function is continuous at \(\mathbf{X}_{0}\), we can conclude that the product function \(G(\mathbf{X}) = h(\mathbf{X}) \mathbf{F}(\mathbf{X})\) is continuous at \(\mathbf{X}_{0}\). Therefore, the product \(h \mathbf{F}=(h f_1, h f_2, \ldots, h f_m)\) is continuous at \(\mathbf{X}_{\mathbf{0}}\).

Key concepts

Multivariable Functions

Multivariable functions are a fascinating topic in calculus. They involve functions that have more than one input variable. For example, a function might be of the form \(f(x, y)\), which means it takes two arguments and returns a value. Similarly, if a function \(\mathbf{F}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\), it means the function takes \(n\) variables as input and outputs \(m\) variables.
Multivariable functions are widely used in fields like physics and engineering, where systems naturally depend on multiple parameters.
Understanding how these functions behave, especially in terms of continuity and differentiability, is crucial in real-world applications.
In essence, a multivariable function must be smooth through all points in its domain to be considered continuous, much like a single-variable function must. This seamless transition across values is what allows predictions and modeling in multivariate systems.

Product of Continuous Functions

The product of continuous functions is a concept that helps in understanding complex behaviors of functions. If you have two continuous functions, say \(f(x)\) and \(g(x)\), then their product \(f(x) \cdot g(x)\) is also continuous. This is a fundamental property of continuous functions.
In the context of our exercise, we have two functions, \(h\) and \(\mathbf{F}\), which when multiplied create a new function \(G(\mathbf{X}) = h(\mathbf{X}) \mathbf{F}(\mathbf{X})\).

• The continuity of \(h\) at a point ensures its behavior can be predicted close to that point.
• Similarly, the continuity of each component of \(\mathbf{F}\) at a point allows for smooth predictions.

Therefore, the product of these functions \(G(\mathbf{X})\) remains continuous. This property is vital in calculus and real analysis, ensuring that interactions between functions don't introduce unexpected behaviors.

Real Analysis

Real Analysis is a branch of mathematical analysis dealing with real numbers and real-valued sequences and functions. It provides a rigorous foundation for calculus and explores concepts like limits, continuity, differentiation, and integration.
Continuity in real analysis is pivotal because it ensures the smooth behavior of functions. A continuous function won’t have abrupt changes or jumps, which makes them easier to handle mathematically.
A function \(f(x)\) is continuous at a point \(x_0\) if, intuitively, small changes in \(x\) near \(x_0\) result in small changes in \(f(x)\). This translates into the formal definition using limits:
\[ \lim_{{x \to x_0}} f(x) = f(x_0) \]

• This concept extends naturally into multivariable calculus, where understanding continuity becomes essential for working with more than one variable.
• In practical applications within real analysis, continuity allows for establishing convergence of sequences and the existence of limits, both foundational for more advanced studies.

This consistency across functional behavior is significant, giving real analysis its robust framework.