Moment generating function of $f(x)=1/c$, $0<x<c$.
Well, I know the calculation is easy. $M_X(t) = \frac{1}{ct} (e^{tc}-1)$.
This question is Exercise 2.30 (a) Statistical Inference Book by George Casella and Roger Lee Berger. However, by definition, we need to discuss the existence of $M_X(t)$. We need to guarantee that the expectation $E e^{tX}$ exists for t in some neighborhood of 0. Obviously, in this question, t cannot be 0. Because if t=0, then mgf doesn't exist. Am I correct?