Convenient S-shaped production function (i.e. with IRS and DRS) to derive a discontinuous demand for labor
I'm not sure but it seems to me that the logistic function $\frac{e^{x}}{1+e^{x}}$ could serve your purpose. 

You may need to scale it as its output falls between 0 and 1, but it does have an analytical derivative that you can then use to solve for the labour demand function.