Finite difference method having a discontinuity

You're approaching this the wrong way by using the product rule of differentiation. Rather, making use of the fact that $$ frac{partial u(x_i)}{partial x} approx frac{u_{i+1/2}-u_{i-1/2}}{Delta x}, $$ the first step in finding what you need to do is to recognize that $$ z(x_i)frac{partial u(x_i)}{partial x} approx z_i frac{u_{i+1/2}-u_{i-1/2}}{Delta x} = frac{z_i u_{i+1/2}-z_i u_{i-1/2}}{Delta x}. $$

The second step is the applying the finite difference operation one more time: $$ frac{partial}{partial x} left[z(x_i)frac{partial u(x_i)}{partial x} right] approx frac{frac{z_{i+1/2} u_{i+ 1}-z_{i+1/2} u_{i}}{Delta x} - frac{z_{i-1/2} u_{i}-z_{i-1/2} u_{i-1}}{Delta x}}{Delta x}. $$ This can be simplified to the following, more familiar form: $$ frac{partial}{partial x} left[z(x_i)frac{partial u(x_i)}{partial x} right] approx frac{z_{i+1/2} u_{i+ 1}-(z_{i+1/2} +z_{i-1/2}) u_{i} + z_{i-1/2} u_{i-1}}{Delta x^2}. $$