structural:expansion, stylistic:style_transfer
Mathematical discussion

A simple criterion for checking if a given stationary point of a real-valued function F (x, y) of two real variables is a saddle point is to compute the function's Hessian matrix at that point: if the Hessian is indefinite, then that point is a saddle point. For example, the Hessian matrix of the function z = x 2 − y 2 {\displaystyle z=x^{2}-y^{2}} at the stationary point (x, y, z) = (0, 0, 0) {\displaystyle (x,y,z)=(0,0,0)} is the matrix

which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point (0, 0, 0) {\displaystyle (0,0,0)} is a saddle point for the function z = x 4 − y 4, {\displaystyle z=x^{4}-y^{4},} but the Hessian matrix of this function at the origin is the null matrix, which is not indefinite.

In the most general terms, a saddle point for a smooth function (whose graph is a curve, surface or hypersurface) is a stationary point such that the curve/surface/etc. in the neighborhood of that point is not entirely on any side of the tangent space at that point.

In a domain of one dimension, a saddle point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum.