Given a coherent sheaf F over a noetherian scheme Y, a classical result in algebraic geometry states the upper-semocontinuity of the function sending any point $y \in Y$ to $\mathrm{dim}_{k(y)}(F \otimes k(y))$.
What about similar results for the length function, i.e. $\mathrm{lt}_{O_y}(F_y)$?