Semicontinuity of length for coherent sheaves

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. $$\mathrm{lt}_{O_y}(F_y)$$?