Interpolation in Sobolev spaces

Hi, I am confused about the pointwise properties of Sobolev functions.
Does an interpolation constraint like f(x_i)=y_i make sense in Sobolev
spaces?
I guess not, but I can't construct two functions in a Sobolev space
which are not pointwise equivalent but the same w.r.t. the Sobolev
space i.e. the norm of their difference is zero.
Does the set on which the Sobolev space is defined play a role?

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