Example of uniform continuous unbounded function such $\lim_{x \to \infty}f(x) \neq \infty$ or $\lim_{x \to \infty}f(x) \neq -\infty$

Question

I wanted to find Example of uniform continuous unbounded function such $\lim_{x \to \infty}f(x) \neq \infty$ or $\lim_{x \to \infty}f(x) \neq -\infty$.But I did not get.I think there must be some function like oscillation with higher wavelenth as x goes large.Any Help will be appreciated

Answer 1

Something like $f(x)=\sqrt x\sin\sqrt x$? From $$f'(x)=\underbrace{\frac{\sin\sqrt x}{2\sqrt x}}_{|\cdot|\le \frac12}+\underbrace{\frac{\cos\sqrt x}{2}}_{|\cdot|\le \frac12},$$ we see that $f$ is Lipschitz with constant $1$, hence uniformly continuous.

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Or (also as a function $[0,\infty)\to\Bbb R$)$$f(x)=\min\{x-\lfloor \sqrt x\rfloor^2-\tfrac12\lfloor\sqrt x\rfloor ,\lceil\sqrt x\rceil^2-x-\tfrac12\lceil\sqrt x\rceil\}$$ Note that $f$ is the piecewise linear interpolation between the points $(k^2,-\frac12k)$ and $(k^2+k+\frac12,\frac12k+\frac12)$, hence again is Lipschitz with constant $1$, hence uniformly continuous.