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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 :)

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# Saddle points of functions of n variables

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