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4) If X is a topological space and ffn X→ R are continuous. Show that both min fi and max1 fi, defined by the equalities (1), are continuous.

4) If X is a topological space and ffn X→ R are continuous. Show that both min fi and max1 fi, defined by the equalities (1), are continuous.

Advanced Engineering Mathematics
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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**Exercise 4: Continuity in Topological Spaces** If \( X \) is a topological space and \( f_1, \ldots, f_n : X \to \mathbb{R} \) are continuous functions, demonstrate that both \( \min_{i=1}^{n} f_i \) and \( \max_{i=1}^{n} f_i \), defined by the equalities (1), are continuous. --- **Explanation**: This exercise involves working with continuous functions within the context of a topological space. Given a set of continuous functions \( f_1, \ldots, f_n \), you are asked to prove that the pointwise minimum and maximum of these functions are also continuous. There are no graphs or diagrams in the image, only mathematical text.
Transcribed Image Text:**Exercise 4: Continuity in Topological Spaces** If \( X \) is a topological space and \( f_1, \ldots, f_n : X \to \mathbb{R} \) are continuous functions, demonstrate that both \( \min_{i=1}^{n} f_i \) and \( \max_{i=1}^{n} f_i \), defined by the equalities (1), are continuous. --- **Explanation**: This exercise involves working with continuous functions within the context of a topological space. Given a set of continuous functions \( f_1, \ldots, f_n \), you are asked to prove that the pointwise minimum and maximum of these functions are also continuous. There are no graphs or diagrams in the image, only mathematical text.
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