general topology - Finding a mapping that is not continuous but open

Find example of a mapping that is open but not continuous and a mapping that is closed but not continuous.

I striving with these questions.

I thought of using for the first case a map between $(\mathbb{Z},\tau)$ where $\tau$ is the co-finite topology to $(\mathbb{R},\tau')$ with the standard topology. I think the mapping would not be continuous and since any open set in $\mathbb{R}$ is infinite and uncountable and the inverse image would need to be countable.
However I am not seeing how to prove this with more rigorous mathematical terminology and I do not see how this mapping might be open.

Question:

Can someone help me solve this question?

Thanks in advance!