Answer
Please see below.
Work Step by Step
The function has nonremovable discontinuities at only even integers since the greatest integer function is continuous everywhere except at integers, so we can find the discontinuities as follows.$$\frac{t+2}{2} = n, \, n \in \mathbb{Z} \quad \Rightarrow \quad t=2(n-1)=2k, \, k \in \mathbb{Z}$$To find how often the company replenishes its inventory, we should find the period of the function by setting $N(t_1)=N(t_2)$, for distinct variables $t_1$ and $t_2$, as follows.$$N(t_1)=N_(t_2) \quad \Rightarrow \quad 25 \left (2 \left [ \frac{t_1+2}{2} \right ] -t_1 \right )=25 \left (2 \left [ \frac{t_2+2}{2} \right ] -t_2 \right ) \quad \Rightarrow \quad t_2-t_1= 2 \left ( \left [ \frac{t_2+2}{2} \right ] - \left [ \frac{t_1+2}{2} \right ] \right )$$The term in the parentheses on the right hand side of the last equation is a subtraction of two integers and so can be any integer, so the least possible absolute value for such term is $1$. Hence, we have$$t_2-t_1=2(1)=2.$$Thus, the company replenishes its inventory every $2$ months.
