Exercise 4. Suppose f: I→ X is a continuous map from the interval ICR to the topological space X. Consider the graph Graph(f) CIX X of f, defined as usual by Graph(f) = {(t, f(t)) | t € I} CR × X}. TORF b) Show that both Graph(f) and its closure Graph(f) in Rx X are connected. PAN
Exercise 4. Suppose f: I→ X is a continuous map from the interval ICR to the topological space X. Consider the graph Graph(f) CIX X of f, defined as usual by Graph(f) = {(t, f(t)) | t € I} CR × X}. TORF b) Show that both Graph(f) and its closure Graph(f) in Rx X are connected. PAN
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
4b
![**Exercise 4.** Suppose \( f: I \rightarrow X \) is a continuous map from the interval \( I \subseteq \mathbb{R} \) to the topological space \( X \). Consider the graph \( \text{Graph}(f) \subseteq I \times X \) of \( f \), defined as usual by
\[
\text{Graph}(f) = \{(t, f(t)) \mid t \in I\} \subseteq \mathbb{R} \times X
\]
b) Show that both \( \text{Graph}(f) \) and its closure \( \overline{\text{Graph}(f)} \) in \( \mathbb{R} \times X \) are connected.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa68164dd-6bba-4aa5-92bc-4824a71db092%2F612b20f8-36da-49e2-b42e-429d8425f939%2Fexa7ex5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 4.** Suppose \( f: I \rightarrow X \) is a continuous map from the interval \( I \subseteq \mathbb{R} \) to the topological space \( X \). Consider the graph \( \text{Graph}(f) \subseteq I \times X \) of \( f \), defined as usual by
\[
\text{Graph}(f) = \{(t, f(t)) \mid t \in I\} \subseteq \mathbb{R} \times X
\]
b) Show that both \( \text{Graph}(f) \) and its closure \( \overline{\text{Graph}(f)} \) in \( \mathbb{R} \times X \) are connected.
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