Let $f : \Bbb R^3 \longrightarrow \Bbb R^3$ be given by $$f(x_1,x_2,x_3) = (e^{x_2} \cos x_1, e^{x_2} \sin x_1, 2x_1 - \cos x_3).$$ Consider a set $$\begin{align*} E & = \left \{(x_1,x_2,x_3) \in \Bbb R^3\ \bigg |\ \text {there exists an open subset}\ U\ \text {around}\ (x_1,x_2,x_3)\ \text {such that}\ f\vert_U\ \text {is an open map} \right \}. \end{align*}$$ Then which of the following are true?
$(1)$ $E = \Bbb R^3$
$(2)$ $E$ is countable
$(3)$ $E$ is not countable but not $\Bbb R^3$
$(4)$ $\left \{\left (x_1,x_2, \dfrac {\pi} {2} \right )\ \bigg |\ x_1,x_2 \in \Bbb R \right \}$ is a proper subset of $E$
How do I proceed? Can anybody please give me some hint?
Thanks for your time.
Source $:$ CSIR NET JUNE $2019.$