Poincaré–Miranda theorem


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

Intuitive description

The picture on the right shows an illustration of the Poincaré–Miranda theorem for functions. Consider a couple of functions whose domain of definition is the square. The function is negative on the left boundary and positive on the right boundary, while the function is negative on the lower boundary and positive on the upper boundary. When we go from left to right along any path, we must go through a point in which is. Therefore, there must be a "wall" separating the left from the right, along which is . Similarly, there must be a "wall" separating the top from the bottom, along which is . These walls must intersect in a point in which both functions are .

Generalizations

The simplest generalization, as a matter of fact a corollary, of this theorem is the following one.
For every variable, let be any value in the range
Then there is a point in the unit cube in which for all :
The this statement can be reduced to the original one by a simple translation of axes,
where