The result should look like this:

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These vertex normals are calculated by averaging the face normals of each connected face. The problem we get when detaching or deleting faces (which we will inevitably do when we go to split up our mesh), is that Max now recalculates the vertex normals based on the faces around it. As we no longer have faces on one side (since we've detached them in this instance), the vertex normals are changed to reflect this. Here's an example of what happens:

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So as you can see, this is the crux of the matter. If we delete/detach faces and the vertex normals get recalculated, we've got to find a way of either keeping those normals, or just simply not deleting those faces.