Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous and monotonically increasing (or decreasing). $f(0)=0$. Does $f$ necessarily attain all possible values in $\mathbb{R}$? i.e. is $f$ onto?
2026-03-30 00:16:47.1774829807
Is a continuous monotone function with domain $\mathbb{R}$ necessarily unbounded?
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Let $f(x) = \arctan(x)$. That should do it, no?