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 $U = f^{-1}(B(f(x), 1))$.

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

Because $f(U)\subset B(f(x), 1)$, $f$ is bounded on $U$.

Thus $f$ is locally bounded.