Is a continuous function on a bounded set bounded itself?

real-analysis continuity

If $f\colon A\to\mathbb R$ is continuous and $A$ is a bounded set, does it necessarily follow that $f$ is bounded?


Consider $f(x) = \dfrac1x$ on $A = (0,1)$.


The answer is No.

But

If a function f is uniformly continuous, then f is bounded on A also.

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