The complex function,

f(z)=z

Can also be expressed in terms of real functions,

f(x+iy)=x+iy

We note the functions x,y are differenciable (Good).

Next we check Cauchy-Riemann Equations:

The partial derivative of x along x is 1

The partail derivative of y along y is 1

So we have,

1=1 (Good)

Next we check the second equation,

The partial derivative of x along y zero,

The negative partial derivative of y along x is zero,

So we have,

0=0 (Good)

Thus, the function,

f(z)=z is homolophric on the entire complex plane. I do not see how you can say it is not.