# Local

A property is called local if it relates only to things very close to the point of interest.

For example, the function $f(x)$ has a local minimum at $x_0$ if $f(x_0)\le f(x)$ whenever $x$ is very close to $x_0$. But there could be values of $x$ with $f(x)<f(x_0)$ when $x$ is further away, as shown in this sketch.

Here, $f(x)=x^3-3x$ has a local minimum at $x=1$, as the point $(1,-2)$ is lower than all the points nearby. However, there are other points on the graph which are lower, such as $(-3,-18)$, so $f(x)$ does not have a global minimum at $x=1$.

A local maximum is defined similarly.