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) $.

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Description: 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 \(

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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).

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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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) $.

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