Say you have a function f(q(x)) where q quantizes x into one of s_1, ..., s_n. Then if q(x) = s_i for a certain x, df/ds_i = df/dq and df/ds_j = 0 for all j != i.

That breaks down for values of x precisely at the boundary between steps, so I should have qualified "differentiable" with "almost everywhere".

It also occurs to me that this might interact strangely with the approximation dq/dx = 1, but since the quantization steps are globally shared, I think it should be stable anyway.

If the evaluation suite for your code doesn't require too much manual interaction, I might try and see for myself.