Continuous Implies Locally Bounded

Theorem

Let $X$ be a topological space.

Let $M$ be a metric space.

Let $f: X \to M$ be continuous.

Then $f$ is locally bounded.

Proof

Let $x \in X$.

Let $U = \map {f^{-1} } {\map B {\map f x, 1} }$.

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By continuity, $U$ is a neighborhood of $x$.

Because $\map f U \subset \map B {\map f x, 1}$, $f$ is bounded on $U$.

Thus $f$ is locally bounded.

$\blacksquare$