The theorem assumes the function has identical domain and codomain. Eg f(x) = x^3 maps [0, 1] to itself.
So that's why they gave the glass stirring example, the domain here is the whole volume of water. So as long as it ends up in the same volume of water, the process of stirring is assumed to be continuous and therefore this theorem applies. Your example changes where it ends up (ie not back to the same domain) and so it cannot be applied.
So that's why they gave the glass stirring example, the domain here is the whole volume of water. So as long as it ends up in the same volume of water, the process of stirring is assumed to be continuous and therefore this theorem applies. Your example changes where it ends up (ie not back to the same domain) and so it cannot be applied.