Every continuous real-valued function on a closed subspace of a normal space extends continuously to the whole space. If its values lie in a closed interval, the extension can be kept in that interval.
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The Tietze Extension Theorem is a fundamental result in topology, particularly in the context of normal spaces. It states that if \( X \) is a normal topological space and \( A \) is a closed subset of \( X \), then any continuous function \( f: A \to \mathbb{R} \) can be extended to a continuous function \( F: X \to \mathbb{R} \).