Do critical points have to be in the domain of f(x,y)?

[Idealistic]

In my notes, I have a critical point occurs when fx(xo,yo) = 0 and fy(y0,x0) = 0 or when either fx(xo,yo) and fx(xo,yo) do not exist.

What if fx(x0,y0) and fy(xo,yo) do not exist, but f(xo,yo) is not in the domain. is (xo,yo) still a critical point?

Bearing in mind this is in R3.

 

[BigGlenntheHeavy]

I went back and forth with Subhotosh Khan on this one concerning a function of one variable.

Same thing applies to a function of two variables, to wit:

Definition of critical point:

Let f be defined (note defined) on an open region R containing (xo,yo) (note containing (xo,yo)). We call (xo,yo) a critical point of f if one of the following is true.

\(\displaystyle 1. \ f_x(x_0,y_0) \ = \ 0 \ and \ f_y(x_0,y_0) \ = \ 0.\)

\(\displaystyle 2. \ f_x(x_0,y_0) \ or \ f_y(x_0,y_0) \ does \ not \ exist.\)

Note; If the suppose point is not in the domain of f, then it doesn't exist for f, hence how can it be a critical point of f? Just plain common horse sense.

 

[Idealistic]

I figured that, Just wanted to be completely certain.

Thanks again.