Saddle points of functions of n variables

[D_Tr]

Hi,

I need some clarification on exactly which critical points are saddle points. The definition I am finding everywhere is that "A saddle point is a stationary point that is neither a local minimum nor a local maximum." My question is: what kind of minimum and maximum is the above definition about? The strict or the non-strict kind?

In case that strict extrema are implied, then, for example, the constant function f(x, y) = c has a saddle point at each point of its domain, whereas in the case of non-strict extrema, the same function has no saddle points, because at every point it has both a local non-strict minimum and a local non-strict maximum. A similar ambiguity arises in the case of, for example, the function f(x, y) = x^2. Are all points that lie on the x = 0 line saddle points since they are non-strict extrema?

Thanks for reading :)

 

[the_wolfman]

As I understand it, a saddle point is a critical (stationary) point that is not a local (non-strict) extreme point.
So you are correct. In your first example, all points in your domain are saddle points. In your second example all points along the line x=0 are saddle points.

 

[D_Tr]

Thanks :) After a bit more googling I found https://www.math.okstate.edu/~atovstolis/lecture_notes/LN14_7.pdf lecture notes where first a distinction is drawn between strict and non-strict extrema and then a saddle point is defined as you too understand it. So I guess my question is answered!