Stolen Necklace problem - Borsuk Ulam - function continuity

Question

This question is based on 3blue1brown's youtube video titled Sneaky Topology | The Borsuk-Ulam theorem and stolen necklaces

The video

• sets up the Stolen Necklace problem
• Explains and proves the Borsuk-Ulam theorem
• Explains how Borsuk Ulam theorem can be used to prove that a split of the necklace is possible under the given constraints

My question is as follows: Borsuk-Ulam has a "continuity" constraint on the function mapping the nd sphere to the n-1d plane. Whereas, in the video, Grant talks about a function

that takes in a necklace allocation and splits out 2 numbers - the number of sapphires and diamonds

It seems to me that this function is not continuous and we cannot use Borsuk-Ulam in this case. Am I misunderstanding something?

Answer 1

Suppose the necklace $N$ is first blue, then green, then blue, then green:

green         __     __
blue  ________  _____
      0               1

This function is discontinuous at the points where there is a change of color. We do not apply Borsuk-Ulam to it. We use a different function.

Suppose we divide the necklace into two parts $[0, a]$ and $[a, 1]$. The function will take $a$ and return how much green is there in $[0,a]$.

For $N$, this is

                      /
                _____/
               /
      ________/
      0               1

This function grows linearly at the intervals where the color is green, and stays constant otherwise. It is continuous. It is effectively an integral of the previous (discontinuous) function.

In the video it's more complicated, because we divide the necklace into three (or more) parts, and the function takes a pair of numbers (or more).