The functional boxplot is perhaps the best-known visualization tool for functional data. We revisit the definition of the functional boxplot and demonstrate that its classical construction, when used in conjunction with the modified band depth for functional data, does not result in a well-defined boxplot. As a remedy to this problem, we argue that the only functional depths that induce well-behaved functional boxplots are those of infimal type. Several practical issues connected with infimal depth-based boxplots are discussed, such as the problem of possible degeneracy and difficulties with identifying shape outlying functions.