Saddle point

Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}$ be differentiable at $a \in U$. The point $a$ is a saddle point of $f$ if $\nabla f(a) = 0$ and $a$ is not a local extremum of $f$.

Remarks

Equivalently, $\nabla f(a) = 0$ and each ball around $a$ contains points $x$ and $y$ with $f(x) < f(a) < f(y)$. The origin is a saddle point of $f(x, y) = x^2 - y^2$. Along the $x$-axis $f$ has a minimum at $0$, and along the $y$-axis it has a maximum. If $f$ is of class $C^2$ and the Hessian matrix at a critical point is indefinite, the point is a saddle point. This condition is not necessary: $f(x, y) = x^3$ has a saddle point at $(0, 0)$ with zero Hessian. In one variable, a saddle point is a stationary point that is not an extremum, such as $0$ for $x \mapsto x^3$.

Explanations

L1 human

The name comes from the shape of a horse saddle. It curves up from front to back and curves down from side to side. So the centre is flat, but it is neither a top nor a bottom.