On a discrete version of the antipodal theorem

The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping ${\displaystyle f:S^k\to {ℝ}^k}$ there exists a point ${\displaystyle x\in S^k}$ such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which ${\displaystyle S^k}$ is replaced by the set of vertices of a high-dimensional cube equipped with Hamming's metric. In place of equality we obtain some optimal estimates of ${\displaystyle \in f_x\left||f\left(x\right)-f(-x)|\right|}$ which were previously known (as far as the author knows) only for f linear (cf. [1]).