Question

Show that the following definitions are equivalent. (a) \(\mathbf{F}=\left(f_{1}, f_{2}, \ldots, f_{m}\right)\) is continuous at \(\mathbf{X}_{0}\) if \(f_{1}, f_{2}, \ldots, f_{m}\) are continuous at \(\mathbf{X}_{0}\). (b) \(\mathbf{F}\) is continuous at \(\mathbf{X}_{0}\) if for every \(\epsilon>0\) there is a \(\delta>0\) such that \(\mid \mathbf{F}(\mathbf{X})-\) \(\mathbf{F}\left(\mathbf{X}_{0}\right) \mid<\epsilon\) if \(\left|\mathbf{X}-\mathbf{X}_{0}\right|<\delta\) and \(\mathbf{X} \in D_{\mathbf{F}}\).

Short answer

The equivalence between the two definitions of continuity for a vector valued function states that if all component functions are continuous at a point (Definition (a)), then the vector valued function also satisfies the epsilon-delta definition of continuity (Definition (b)), and vice versa. This means that both definitions essentially express the same concept of continuity for vector valued functions.

Step-by-step solution

Prove that definition (a) implies definition (b)

Given that all the components of the function are continuous, we know that for each \(f_i\) and for any \(\epsilon_{i}>0\), there exists a \(\delta_{i}>0\) such that \(|f_i(\mathbf{X})-f_i(\mathbf{X}_0)|<\epsilon_{i}\), whenever \(|\mathbf{X}-\mathbf{X}_0|<\delta_{i}\). So, let's choose \(\frac{\epsilon}{\sqrt{m}}\) for each \(i\). Now from the component-wise condition, for each \(i\), there is a \(\delta_{i}>0\) such that when \(|\mathbf{X}-\mathbf{X}_0|<\delta_{i}\), we have \(|f_i(\mathbf{X})-f_i(\mathbf{X}_0)|<\frac{\epsilon}{\sqrt{m}}\). Let \(\delta = \min \{\delta_1, \delta_2, \ldots, \delta_m\}\). Now, whenever \(|\mathbf{X}-\mathbf{X}_0|<\delta\), we have $$\mid \mathbf{F}(\mathbf{X})-\mathbf{F}\left(\mathbf{X}_{0}\right) \mid = \sqrt{|f_1(\mathbf{X})-f_1(\mathbf{X}_0)|^2 + |f_2(\mathbf{X})-f_2(\mathbf{X}_0)|^2 + \cdots + |f_m(\mathbf{X})-f_m(\mathbf{X}_0)|^2}$$ $$< \sqrt{\left(\frac{\epsilon}{\sqrt{m}}\right)^2 + \left(\frac{\epsilon}{\sqrt{m}}\right)^2 + \cdots + \left(\frac{\epsilon}{\sqrt{m}}\right)^2} = \sqrt{m\left(\frac{\epsilon}{\sqrt{m}}\right)^2} = \epsilon$$ So, definition (a) implies definition (b).

Prove that definition (b) implies definition (a)

We are given that for every \(\epsilon>0\), there is a \(\delta>0\) such that \(\mid \mathbf{F}(\mathbf{X})-\mathbf{F}\left(\mathbf{X}_{0}\right) \mid<\epsilon\) if \(|\mathbf{X}-\mathbf{X}_{0}|<\delta\) and \(\mathbf{X} \in D_{\mathbf{F}}\). Now, for every \(\epsilon>0\), we can have $$\mid \mathbf{F}(\mathbf{X})-\mathbf{F}\left(\mathbf{X}_{0}\right) \mid = \sqrt{|f_1(\mathbf{X})-f_1(\mathbf{X}_0)|^2 + |f_2(\mathbf{X})-f_2(\mathbf{X}_0)|^2 + \cdots + |f_m(\mathbf{X})-f_m(\mathbf{X}_0)|^2}<\epsilon$$ Since the above inequality holds for all \(\epsilon>0\), we can also say that for each component function \(f_i(\mathbf{X})\), there exists a \(\delta_{i}>0\) such that \(|f_i(\mathbf{X})-f_i(\mathbf{X}_{0})|<\epsilon\) whenever \(|\mathbf{X}-\mathbf{X}_{0}|<\delta_{i}\), which means that each component function \(f_i\) is continuous at \(\mathbf{X}_0\). Thus, definition (b) implies definition (a). We have proved the equivalence of the given definitions.

Key concepts

Epsilon-Delta Definition of Continuity

In real analysis, the epsilon-delta definition of continuity is a formal way to describe what it means for a function to be continuous at a point. This definition is fundamental in mathematics because it sets a precise foundation for continuity, allowing us to explore and analyze functions rigorously.

According to this definition, a function \( f \) is continuous at a point \( x_0 \) if for every positive number \( \epsilon \), there exists a positive number \( \delta \) such that whenever the distance between \( x \) and \( x_0 \) is less than \( \delta \), the distance between \( f(x) \) and \( f(x_0) \) is less than \( \epsilon \).

This can be symbolically written as:

• For every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( |x - x_0| < \delta \), then \( |f(x) - f(x_0)| < \epsilon \).

In the case of multivariable functions, such as \( \mathbf{F} \) in our exercise, this concept is similarly applied but takes into account the vector nature of these functions, leading to a combination of the distances between function outputs and inputs across all dimensions.

Component-wise Continuity

Component-wise continuity is an essential concept when dealing with vector-valued functions, like the function \( \mathbf{F} = (f_1, f_2, \ldots, f_m) \). This approach simplifies the study of multivariable functions by checking the continuity of each component function individually.

To understand why component-wise continuity is effective, consider that if each component function \( f_i \) is continuous at a point \( \mathbf{X}_0 \), then the entire vector-valued function \( \mathbf{F} \) is continuous at that point. This is because the behavior of \( \mathbf{F} \) is essentially the combination of all the component functions, evaluated together. The continuity of each part ensures the continuity of the whole.

Practically, this means instead of struggling with the entire function at once, you can examine the simpler part functions independently:

• Check if each \( f_i \) is continuous at \( \mathbf{X}_0 \).
• If they all are, \( \mathbf{F} \) is continuous at \( \mathbf{X}_0 \).

This systematic approach is particularly useful in fields like engineering and physics, where real-world systems are often modeled as multivariable functions.

Real Analysis

Real analysis is a branch of mathematics focused on real numbers and real-valued sequences and functions. It is a crucial area of study that provides the tools and frameworks for understanding concepts such as limits, continuity, derivatives, and integration.

In real analysis, making intuitive concepts precise is key, especially concerning continuity and limits. This is where definitions like epsilon-delta come into play. They allow mathematicians to handle abstract ideas with accuracy and provide a foundation for further work in calculus and advanced mathematics.

The field is extensive and enables the solving of complex problems by:

• Establishing rigorous foundations for calculus.
• Exploring the properties of sequences and series.
• Analyzing the behavior of functions thoroughly.

Whether in pure or applied mathematics, real analysis equips students and professionals with the analytical skills necessary for advanced study and applications in various scientific and engineering domains.