# Thread: Show that a function is not differentiable anywhere...

1. ## Show that a function is not differentiable anywhere...

Show that the function g(z)=Im z is not differentiable anywhere using the definition of f to be differentiable at a point 'z' . where f:u (tending to)Complex numbers.

for f to be differentiable then the limit of
lim [f(z) - f(z0)]/(z-z0) (z tending to z0) must exist. But how so I show that this is not the case with g(z)=Im z ...the imaginary part?

Thanks

2. In order for the complex derivative to exist, the differential quotient must be the same for any sequence approaching z_0.
For g(z)=Im z you can find 2 complex sequences approaching z_0, for which the differential quotient is different.

Take for example
a_n:=z_0+i/n
and b_n:=z_0+1/n