Anybody with elementary mathematical education will have seen improper fractions to refer to fractions where the numerator is greater than or equal to the denominator.
At a certain point in calculus studies, students are confronted with improper integrals, that is the limit of a definite integral as the bound(s) tend to $\infty$ or $-\infty$.
Improper prior distributions, in Bayesian statistics, are prior distributions that don't integrate to $1$.
Personally, as I learned these concepts, I found the use of the word improper to hint to a negative connotation.
A double question here.
$1)$ Are there any other objects in mathematical studies where this adjective is used?
$2)$ Were these mathematical objects named like this simply because they were extensions (one might also say "dares", such as taking the limit of a definite integral) of previously consolidated concepts or was this nomenclature used because of possible discord on the existence of such objects while they were being discovered?