There can be various generalizations. My favorite one is this.

Let $f_1,\ldots,f_n$ be continuous functions on the unit cube. And suppose that each of them take values of opposite sign on the opposite facets (each on its own pair of the opposite facets). Then all $f_j$ have a common zero inside the cube.

There can be various generalizations. My favorite one is this.

Let $f_1,\ldots,f_n$ be continuous functions on the unit cube. And suppose that each of them takes values of opposite sign on the opposite facets (each on its own pair of the opposite facets). Then all $f_j$ have a common zero inside the cube.

There can be various generalizations. My favorite one is this, Let $f_1,\ldots,f_n$ be continuous functions on the unit cube. And suppose that each of them take values of opposite sign on the opposite facets (each on its own pair of the opposite facets). Then all $f_j$ have a common zero inside the cube.

There can be various generalizations. My favorite one is this.

Let $f_1,\ldots,f_n$ be continuous functions on the unit cube. And suppose that each of them take values of opposite sign on the opposite facets (each on its own pair of the opposite facets). Then all $f_j$ have a common zero inside the cube.