Finite evaluation versus global functions

For any prescribed nonempty finite $S=\{t_1,\dots ,t_s\}\subset \mathbb{R}$, let

Then $m{|}_S=n_1{|}_S$, but $m$ is not a constant function. This gives a finite fit without a global identity. The functions are constructed after the sample is prescribed; it is not an assertion that one fixed nonmember fits every finite sample.