Why aren't your fingerprints affected by cuts or injuries to your fingers?

Answer 1

because after you have a cut, the injurie heals.

Answer 2

The skin on your fingers and hands heals incredibly quickly and usually without scarring. Look at your hands, do you see any scars on your fingertips. Even the deepest layers of the skin is replaced quickly on the fingers. The fact that it does take time to heal and that fingerprints aren't exactly the same between your forefinger and thumb (or any two fingers) is why all ten fingers are printed when someone is arrested. Sometimes someone does to enough damage to leave a permanent mark or scar but a quick look at the fingers is enough to know that there is a scar there and the person would be refingerprinted if arrested.