First, you need to be perfectly clear about the topological space in which this whole problem statement is to be situated.

If it is a metric space you can do the following:

1. To show that is closed (for ) you show that the limit of any convergent sequence of functions also lies in .

2. To show that is not closed (for ) you show that there exists a sequence of functions in , the limit of which does not lie in . (For example: it is easy to find a sequence of functions with limit .)