I know that $1/x$ is unbounded on $(0,5)$ (for example) and that since it is unbounded, it is not uniformly continuous.
Does a function have to be bounded to be uniformly continuous? I don't think one exists.
Existence of Unbounded Uniformly Continuous Functions
continuityreal-analysisuniform-continuity
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Best Answer
The function $f(x) = x$ is unbounded on $\mathbb{R}$, but uniformly continuous on $\mathbb{R}$. The function $f(x) = \sqrt{x}$ is another interesting example.
Perhaps you meant to ask something like, if $I$ is a bounded interval (not necessarily closed) and $f: I \to \mathbb{R}$ is uniformly continuous, then is $f$ bounded? The answer to this is yes. Find $\delta > 0$ such that for $|x - y| < \delta$, $|f(x) - f(y)| < 1$. Then by partitioning the interval $I$ up into a finite number of pieces smaller than $\delta$, you can show $f$ is bounded.
The same holds true if $I$ is any bounded set, not just an interval.