The growth of bacteria is given by the function of time ($t$) (in hours) $f(t)=500 e^{0.25 t}$, then find:
a. Initial number of bacteria at $t=0$
b. Number of bacteria after $t=4$ hours
c. Number of bacteria after $t=8$ hours
d. The time if number of bacteria is $500 e^{3}$
e. Is $f(t)$ continuous at $t=4$ hours and $t=8$ hours?