Given the function: f(z) = z/(z-1)

I have made it into the form:

u(x,y) + I * v(x,y)

Caunchy-Riemann equations are only satisfied when y=0 according to my calculations.

Is this correct? Shoudn't the C.R equations hold everywhere except x=1 ? as from the above f(z) there is a singularity at z=1