Give an example of the case or explain why it's not possible for each of the following:

1.) Continuous function $\displaystyle f : (0,1) \rightarrow \mathbb{R}$ and also a Cauchy sequence $\displaystyle (x_n)$ such that $\displaystyle f(x_n)$ is not a Cauchy sequence.

2.) Continuous function $\displaystyle f : [0,1] \rightarrow \mathbb{R}$ and also a Cauchy sequence $\displaystyle (x_n)$ such that $\displaystyle f(x_n)$ is not a Cauchy sequence.

3.) Continuous function $\displaystyle f : [0,\infty) \rightarrow \mathbb{R}$ and also a Cauchy sequence $\displaystyle (x_n)$ such that $\displaystyle f(x_n)$ is not a Cauchy sequence.

4.) Continuous function $\displaystyle f$ that's bounded on $\displaystyle (0,1)$ and that has a maximum value on the open interval, however, does not have a minimum value.