real analysis - Continuous bounded function $f:mathbb{R}rightarrow mathbb{R}$

Question is to check which of the following holds (only one option is correct) for a continuous bounded function $f:\mathbb{R}\rightarrow \mathbb{R}$.

• $f$ has to be uniformly continuous.


• there exists a $x\in \mathbb{R}$ such that $f(x)=x$.


• $f$ can not be increasing.


• $\lim_{x\rightarrow \infty}f(x)$ exists.

What all i have done is :

• $f(x)=\sin(x^3)$ is a continuous function which is bounded by $1$ which is not uniformly continuous.



• suppose $f$ is bounded by $M>0$ then restrict $f: [-M,M]\rightarrow [-M,M]$ this function is bounded ad continuous so has fixed point.


• I could not say much about the third option "$f$ can not be increasing". I think this is also true as for an increasing function $f$ can not be bounded but i am not sure.


• I also believe that $\lim_{x\rightarrow \infty}f(x)$ exists as $f$ is bounded it should have limit at infinity.But then I feel the function can be so fluctuating so limit need not exists. I am not so sure.

So, I am sure second option is correct and fourth option may probably wrong but i am not so sure about third option.

Please help me to clear this.

Thank You. :)

Answer

For the third point, consider $f(x) = \arctan(x)$. For the fourth point, you've already found a counterexample in one of your other points!