---
tag: 00N3T18QNA
kind: definition
name: Saddle point
msc: [26B05]
defines: 0JW6MDHAZD
uses: [04CPG9K1C4, 0NS2QYVC4Y, 0135XZZQ8M]
sources:
  - url: https://en.wikipedia.org/wiki/Saddle_point
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Saddle_point
    locator: Mathematical discussion
---

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $f \colon U \to \mathbb{R}$ be [[differentiable|0135XZZQ8M]] 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|04CPG9K1C4]] 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|0SCSQWXRA4]] 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.