I am wondering if the Maximum Likelihood Estimate exists for a uniform distribution on $[0, \theta)$ (open interval on $\theta$). Choosing $\theta_{MLE}$ as $max\{\bar x\}$ would mean that the $\Pr(\bar x|\theta_{MLE}) = 0$, which obviously isn't the maximum of $\Pr(\bar{x})$.
I found many websites loosely interchanging the open and closed interval during the solution itself so I am not sure if people tend to give any attention to this.