My mum was cleaning a metal unit and she cut her finger her index finger at the side of the nail she said the cut is numb and the cut is black?

Go see a doctor: If it's numb, she may have cut a small nerve in the skin. That may or may not improve after it heals. The worrisome part is that it is black. Go have someone look at it. Maybe the black color is blood under the nail? But it might be infection requiring drainage or antibiotics. It's always good to update the tetanus shot if it's close to being due. They'll ask about that when you go in. Good luck!