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Remember from the definition of derivative that lim delta z goes to 0 must be the same regardless of the manner in which delta z goes to 0.

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This constraint is not really in the definition of the derivative is it?

Oh yes it is!!Look at any classical textbook on the calculus and you will find that the limit not only has to exist, but must be the same

regardless of the direction in which you come at it.

This is fundamental.In the theory of differentiation of functions of a real variable if you are differentiating a function f(x) at x=0, then the limit coming towards x=0 through negative values

must be the same as that obtained by coming through positive values.

That is why y=|x| is not differentiable at x=0 because the limits mentioned above are 1 and -1.

This same idea is carried over into complex functions, so that if the limit is approached through real values or imaginary values or any other way, the

same limit must result in

all cases.

On page 194 Penrose is saying that if the function

phi is to be holomorphic then it cannot be a function of

z bar, but only of

z, and so he sets the derivate of

phi with respect to

z bar to zero and obtains the CR equations.

My proof shows the same thing but only using the basic definition of derivative carried over from functions of a real variable.