Theorem 7.7. LetX be a space with a dense set D, and let Y be Hausdorff. Let f : X → Y and g : X → Y be continuous functions such that for every d in D, f(d) = g(d). Then for all x in X, f(x) = g(x).

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Could you explain how to show 7.7 in detail?

**Definition.** Let \( X \) and \( Y \) be topological spaces. A function or map \( f : X \rightarrow Y \) is a **continuous function** or **continuous map** if and only if for every open set \( U \) in \( Y \), \( f^{-1}(U) \) is open in \( X \).

**Definition.** Let \( f : X \rightarrow Y \) be a function between topological spaces \( X \) and \( Y \), and let \( x \in X \). Then \( f \) is **continuous at the point** \( x \) if and only if for every open set \( V \) containing \( f(x) \), there is an open set \( U \) containing \( x \) such that \( f(U) \subset V \). Thus a function \( f : X \rightarrow Y \) is continuous if and only if it is continuous at each point.

**Theorem 7.7.** Let \( X \) be a space with a dense set \( D \), and let \( Y \) be Hausdorff. Let \( f : X \rightarrow Y \) and \( g : X \rightarrow Y \) be continuous functions such that for every \( d \) in \( D \), \( f(d) = g(d) \). Then for all \( x \) in \( X \), \( f(x) = g(x) \).
Transcribed Image Text:**Definition.** Let \( X \) and \( Y \) be topological spaces. A function or map \( f : X \rightarrow Y \) is a **continuous function** or **continuous map** if and only if for every open set \( U \) in \( Y \), \( f^{-1}(U) \) is open in \( X \). **Definition.** Let \( f : X \rightarrow Y \) be a function between topological spaces \( X \) and \( Y \), and let \( x \in X \). Then \( f \) is **continuous at the point** \( x \) if and only if for every open set \( V \) containing \( f(x) \), there is an open set \( U \) containing \( x \) such that \( f(U) \subset V \). Thus a function \( f : X \rightarrow Y \) is continuous if and only if it is continuous at each point. **Theorem 7.7.** Let \( X \) be a space with a dense set \( D \), and let \( Y \) be Hausdorff. Let \( f : X \rightarrow Y \) and \( g : X \rightarrow Y \) be continuous functions such that for every \( d \) in \( D \), \( f(d) = g(d) \). Then for all \( x \) in \( X \), \( f(x) = g(x) \).
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