(Complex Variables) Differentiability of Arg z

[irony of truth]

I am proving that the function f(z) = Arg z is nowhere differentiable by using the definiton of a derivative. I let z = x + yi. Then, if the limit exists, we have

f'(z) = lim (/\z -> 0) ( f(z + /\z) - f(z) ) / /\z.

(Note that /\ is the triangle symbol)
Also, let /\z = p + iq, where p and q are real values.

Arg z = Tan^-1 (y/x)...how will I continue from here?

 

[relinquished™]

so you want to get

<br /> f&#039;(z) = \lim_{\Delta z \rightarrow 0} \frac{f(z+\Delta z) - f(z)}{\Delta z}<br />

Express the limit in terms of u(x_0,y_0) and v(x_0,y_0), that is,x_0, y_0, \Delta x, \Delta y. then evaluate the limit using 2 approaches: when \Delta x = 0 and \Delta y = 0. If f(z) = Arg z is differentiable, the derivative should be equal in both cases.