$f: \mathbb{R^3} \rightarrow \mathbb{R^2}$ is defined as $$f(x,y,z)= (ze^{[x]}, ze^{[y]}).$$ Is it continuous at $(0,0,0)$ or not? If it is continuous then how can I show it by using epsilon delta definition or if not then how to prove discontinuity. I am trying to show it by the continuity of coordinate functions. I know $ e^{[x]} $ is not continuous at $(0,0,0)$, but I don't know if $ ze^{[x]} $ is continuous or not at $(0,0,0)$.
($[ ]$ is greatest integer function.)
Guide:
Note that if we let $\left\|(x,y,z)\right\|_2 < 1$,
then we have $$\left\|f(x,y,z)\right\|_2^2=z^2\left(\exp(2\lfloor x\rfloor) +\exp (2\lfloor y \rfloor) \right)\le 2z^2$$
that is we have $$\left\|f(x,y,z)\right\|_2 \le \sqrt2|z|$$
Hopefully, you can choose your $\delta$ to solve the problem.