jmc

By definition of $p$, for any prime number $x$ such that $2\le x\le 10$, then $[x+1,x+2)\subset \text{range}(p)$. It follows that $[3,4)\cup [4,5)\cup [6,7)\cup [8,9)\subset \text{range}(p)$. Since the largest prime factor of a composite number less than or equal to $10$ is $5$, then the largest possible value of $p$ on a composite number is $p(10)=p(5)+1=7$. Also, we notice that $[5,6)\subset \text{range}(p)$, since for any $x\in [6,7)$, then $p(x)=p(3)+(x+1-\lfloor x\rfloor )=5+x-\lfloor x\rfloor$. Combining all of this, it follows that the range of $p$ is equal to $[3,5)\cup [6,7)\cup [8,9)\cup \{7\}\cup [5,6)=\boxed{[3,7]\cup [8,9)}$.