What Are Critical Values in Non-Analytic Functions?

[gvk]

There is a problem in Arnold's Trivia :
Find the critical values and critical points of the mapping z -> z^2+ 2*conjugate(z); (sketch the answer).
It seems to me that the critical values (z where f'(z)=0) do not exist, becouse the function f(z)= z^2+ 2*conjugate(z) is not analytic.
Is anybody able to explain what Arnold mean under the notion 'critical values and critical points'?

Thanx for help.

 

[phoenixthoth]

if I'm not mistaken, a critical value would also be a place where f' is undefined (in addition to a point where f' is zero).

you can use the cauchy riemann equations but i think you have to see if the function is analytic at the origin using the limit definition of derivative though I'm not sure.

 

[M98Ranger]

The concept of critical values and critical points is typically used in the context of analytic functions, which are functions that are differentiable at every point in their domain. In this case, it appears that the function given, f(z) = z^2 + 2*conjugate(z), is not analytic, as the poster mentioned. This means that the function may not be differentiable at certain points, and thus the concept of critical values and critical points may not be applicable here. Without further context or clarification from Arnold, it is difficult to determine exactly what he means by these terms in this particular situation. It is possible that he may be using them in a more general sense, but without more information it is impossible to say for sure. It may be helpful to seek clarification from Arnold or to consult with a mathematics expert for a more thorough explanation.