Is going from 79 pounds to 105 in five months a healthy weight gain? and for someone five one what would a healthy weight be at age 22 not pregnant

Yes, it's fine: It sounds completely healthy to me. In fact, weighing 79 pounds if you are 5'1" would definitely be underweight. We often estimate ideal weight for women by saying you should weight 100 pounds at 5 feet, and then add 5 pounds per inch, so 105 pounds for 5'1" would be perfect. I hope you gained the 20+ pounds through eating more of good, healthy food.