Poincaré–Miranda theorem

In mathematics, the Poincaré–Miranda theorem is a generalization of intermediate value theorem, from a single function in a single dimension, to n functions in n dimensions. It says as follows:

Consider ${\displaystyle n}$ continuous, real-valued functions of ${\displaystyle n}$ variables, ${\displaystyle f_{1},\ldots ,f_{n}\colon [-1,1]^{n}\to \mathbb {R} }$. Assume that for each variable ${\displaystyle x_{i}}$, the function ${\displaystyle f_{i}}$ is nonpositive when ${\displaystyle x_{i}=-1}$ and nonnegative when ${\displaystyle x_{i}=1}$. Then there is a point in the ${\displaystyle n}$-dimensional cube ${\displaystyle [-1,1]^{n}}$ in which all functions are simultaneously equal to ${\displaystyle 0}$.

The theorem is named after Henri Poincaré — who conjectured it in 1883 — and Carlo Miranda — who in 1940 showed that it is equivalent to the Brouwer fixed-point theorem. It is sometimes called the Miranda theorem or the Bolzano–Poincaré–Miranda theorem.