If you cut a piece of copper into two pieces, both pieces are still copper. If you continue to cut, you eventually get the smallest possible piece of copper, and it can not be cut. So then copper is an element, and the smallest possible piece is an atom, a Greek word meaning "no cut".
If you cut a piece of sugar into two pieces, both pieces are still sugar. If you continue to cut, you eventually get the smallest possible piece of sugar, and it can be cut but the pieces are not sugar any more. So then sugar is a compound, and the smallest possible piece is a molecule, a Greek word meaning "smallest part".
Paper is an organic product composed of many compounds forming cells which are stuck together with a compound called cellulose, a Greek word meaning "cell sugar". The cellulose is softened in water and the cells are rearranged so you have paper instead of the wood pulp you started with. So when you cut paper you have two pieces of paper, but if you keep cutting it starts to get complicated.
ANSWER: Cutting paper is a mechanical act which does not change the nature of the material. It is still paper.