Are both of the following sentences grammatically correct and, if so, is there any siginificant difference between simple and progressive aspect here?
I give up (on sth. / sb.)
I'm giving up (on sth. / sb.)
Are both of the following sentences grammatically correct and, if so, is there any siginificant difference between simple and progressive aspect here?
I give up (on sth. / sb.)
I'm giving up (on sth. / sb.)
Both of your examples are correct and would be understood as meaning the same thing.
A slight nuance might be
I give up
has more of a finality to it, you are already done with dealing with something, whereas
I'm giving up
may mean you are about to give up but have not quite yet.
The more common of them is "to give up" since it is mostly a quick non-continuous action. However, "to be giving up" can often be used especially to mean that the action is taking too long to complete.
Much depends on the context you are using the verb in.
As for "to give up on" versus "to be giving up on" the latter conveys a certain degree of intention whereas the former states a completed action.
Purely my thought is that mostly the latter can be undone. There is a certain percent of "unless" coming in action whereupon "to be giving up on" changes to the opposite. With "to give up on", this percent is barely there.