What is the function of a median nerve?

It also supplies the muscles of the thenar eminence by a recurrent thenar branch. The rest of the intrinsic muscles of the hand are supplied by the ulnar nerve. The median nerve innervates the skin of the palmar side of the thumb, the index and middle finger, half the ring finger, and the nail bed of these fingers.this should be the answer to your question 

The median nerve innervates the skin of the palmar side of the thumb, the index and middle finger, half the ring finger, and the nail bed of these fingers.