Asked to the same effect, What invertible functions monotonically approach positive andor negative infinity while same is not true over full domain of its first derivative?
Without the bijectivity requirements, non-damped harmonics could count. Without the "differentiably monotonically increasing" requirement (c.f. "monotonically increasing" or merely "increasing") and possibly also the other requirement, trigonometric functions defined over a one-period interval yielding two vertical asymptotes (instead non-endbehavior asymptotes if domain not restricted) could count.
I would prefer solution that does not restrict domain by a manufactured parameter and is more "elementary" in fashion (c.f., e.g. stochastics, multivariable equations, or more clear such as elliptic integrals), but am interested also in forms not satisfying this preference; or better yet an answer of multiple if not all all families that are of, or yielding of some, potential solutions satisfying the necessary conditions.
I am also interested in any cases that might exist of being [once, twice, nth]-differentiably monotonic but not infinitely-differentiably monotonic, as well existene or non- functions that aren't monotonic over its whole domain but in which the requested conditions still to the endbehavior portion(s) of its domain (i.e. past its last "turning point") do apply (satisfying as many of the original restrictions as possible).
My motivating inspiration for this question is in considering and pursuing answer to another question concerning differing scales of infinities seen as end-behavior ratioes (e.g. infinitive iteration of multiplication versus addition with same starting_index and argument, or infinitively repeat-nested function that is exponential versus polynomial).