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Chapter 5: Problem 29

Give an example of a connected set in \(\mathbb{R}^{n}\) that is not polygonally connected.

Short Answer

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Question: Provide an example of a connected set in \(\mathbb{R}^{n}\) that is not polygonally connected. Answer: One example of such a set is given by \(S \subset \mathbb{R}^2\) defined as: $$ S = \{(x, y) \in \mathbb{R}^2 : y = e^x \text{ or } x = 0, y \ge 1\} $$ This set consists of the graph of the exponential function \(y = e^x\) along with the vertical line segment (half line) at \(x = 0\) and \(y \ge 1\). The set S is connected as it consists of continuous functions \(f(x) = e^x\) and \(g(x) = x\), but it is not polygonally connected, as there is no polygonal path between points like \(A=(0,1)\) and \(B=(1, e)\) within the set S.

Step by step solution

01

Define the set

Let's consider the set \(S \subset \mathbb{R}^2\) defined as: $$ S = \{(x, y) \in \mathbb{R}^2 : y = e^x \text{ or } x = 0, y \ge 1\} $$ This set consists of the graph of the exponential function \(y = e^x\) along with the vertical line segment (half line) at \(x = 0\) and \(y \ge 1\).
02

Show that the set is connected

To show that S is connected, we consider the continuous functions \(f(x) = e^x\) and \(g(x) = x\). For any point \((x, e^x) \in S\) on the exponential curve, there exists another point \((0, x)\) in S such that \(|f(x) - g(x)| = |e^x - x|\). Since both f(x) and g(x) are continuous functions over \(\mathbb{R}\), their difference is also continuous. Therefore, S is a connected set.
03

Show that the set is not polygonally connected

To show that S is not polygonally connected, we consider two points, \(A=(0,1)\) and \(B=(1, e)\). A polygonal path between A and B consists of a finite number of line segments, but because the exponential function \(y = e^x\) is strictly increasing and never intersects the vertical line \(x = 0\) for \(y \ge 1\), there cannot be any such polygonal path between A and B within the set S. Therefore, the set S is not polygonally connected.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Function
The exponential function, represented as \(y = e^x\), is a unique and important function in mathematics. It is characterized by the constant base \(e\), which is approximately equal to 2.718. This function has several key properties that make it stand out:
  • It is strictly increasing, which means that as \(x\) gets larger, the value of \(e^x\) also increases.
  • Its graph is a smooth, continuous curve that never touches the horizontal axis but approaches it as \(x\) decreases towards negative infinity.
  • It is defined for all real numbers, and the range is always positive, stretching from \(0\) to \(\infty\).
These properties are crucial when considering connectivity, particularly in the set \(S\) mentioned in the exercise. The exponential curve in \(S\) contributes to the overall connectedness of the set, even though it does not permit polygonal connections due to its strictly increasing nature.
Real Analysis
Real analysis is a branch of mathematics dealing with real numbers and the real-valued functions that arise from them. It provides a rigorous framework for discussing concepts such as limits, continuity, and connectedness, all of which are relevant to the problem at hand.
  • **Limits and Continuity**: Real analysis thoroughly examines functions where limits and continuity play a crucial role. By ensuring functions like \(f(x) = e^x\) and \(g(x) = x\) are continuous, we establish that their combination into set \(S\) doesn't break connection.
  • **Connectedness**: In real analysis, a set is connected if there is no way to partition it into two non-empty open subsets. This property of \(S\) ensures there are no gaps or jumps in the set.
By using these real analysis principles, we can conclude why certain sets maintain connectedness, like \(S\), despite their complicated structures or paths they may not allow.
Path Connectedness
Path connectedness is a more stringent concept of connectivity in mathematics, specifically within the realm of topology. A set is path-connected if any two points within the set can be joined by a continuous path lying entirely within the set.
The distinction between connectedness and path connectedness is subtle yet significant.
  • **Simple Connectivity**: A path-connected set is always connected because if two points can be joined by a path, it implies no breaks exist in the set.
  • **Polygonal Paths**: These are paths made up of straight-line segments. A set that is not polygonally connected, like \(S\), cannot have every path made of these segments without leaving the set.
In the exercise, \(S\) is not polygonally connected due to the strict increase of the exponential part of the set, preventing a finite number of line segments from connecting two points like \((0,1)\) and \((1,e)\) entirely within \(S\). This illustrates how sets can be connected globally while not allowing certain types of paths.

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Most popular questions from this chapter

Calculate \(\partial f(\mathbf{X}) / \partial \Phi\). (a) \(f(x, y)=x^{2}+2 x y \cos x, \quad \Phi=\left(\frac{1}{\sqrt{3}},-\sqrt{\frac{2}{3}}\right)\) (b) \(f(x, y, z)=e^{-x+y^{2}+2 z}, \quad \Phi=\left(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)\) (c) \(f(\mathbf{X})=|\mathbf{X}|^{2}, \quad \Phi=\left(\frac{1}{\sqrt{n}}, \quad \frac{1}{\sqrt{n}}, \cdots, \frac{1}{\sqrt{n}}\right)\) (d) \(f(x, y, z)=\log (1+x+y+z), \quad \boldsymbol{\Phi}=(0,1,0)\)

Let $$ g(\mathbf{X})=\frac{\left(x^{2}+y^{4}\right)^{3}}{1+x^{6} y^{4}} $$ Show that \(\lim _{|x| \rightarrow \infty} g(x, a x)=\infty\) for any real number \(a\). Does $$ \lim _{|\mathbf{X}| \rightarrow \infty} g(\mathbf{X})=\infty \text { ? } $$

Let \(f\) be defined on \(\mathbb{R}^{n}\) by $$f(\mathbf{X})=g\left(x_{1}\right)+g\left(x_{2}\right)+\cdots+g\left(x_{n}\right)$$ where $$g(u)=\left\\{\begin{array}{ll} u^{2} \sin \frac{1}{u}, & u \neq 0, \\ 0, & u=0 . \end{array}\right.$$ Show that \(f\) is differentiable at \((0,0, \ldots, 0)\), but \(f_{x_{1}}, f_{x_{2}}, \ldots, f_{x_{n}}\) are all discontinuous at \((0,0, \ldots, 0)\).

Find all second-order partial derivatives of the following functions at (0,0) . (a) \(f(x, y)=\left\\{\begin{array}{ll}\frac{x y\left(x^{2}-y^{2}\right.}{x^{2}+y^{2}}, & (x, y) \neq(0,0), \\ 0, & (x, y)=(0,0)\end{array}\right.\) (b) \(f(x, y)=\left\\{\begin{array}{ll}x^{2} \tan ^{-1} \frac{y}{x}-y^{2} \tan ^{-1} \frac{x}{y}, & x \neq 0, \quad y \neq 0 \\ 0, & x=0 \quad \text { or } \quad y=0\end{array}\right.\) \(\left(\right.\) Here \(\left.\left|\tan ^{-1} u\right|<\pi / 2 .\right)\)

Find \(\partial f\left(\mathbf{X}_{0}\right) / \partial \Phi,\) where \(\Phi\) is the unit vector in the direction of \(\mathbf{X}_{1}-\mathbf{X}\) ). (a) \(f(x, y, z)=\sin \pi x y z ; \quad \mathbf{X}_{0}=(1,1,-2), \quad \mathbf{X}_{1}=(3,2,-1)\) (b) \(f(x, y, z)=e^{-\left(x^{2}+y^{2}+2 z\right)} ; \quad \mathbf{X}_{0}=(1,0,-1), \quad \mathbf{X}_{1}=(2,0,-1)\) (c) \(f(x, y, z)=\log (1+x+y+z) ; \quad \mathbf{X}_{0}=(1,0,1), \quad \mathbf{X}_{1}=(3,0,-1)\) (d) \(f(\mathbf{X})=|\mathbf{X}|^{4} ; \quad \mathbf{X}_{0}=\mathbf{0}, \quad \mathbf{X}_{1}=(1,1, \ldots, 1)\)
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