using script to fill the uv array

I've a character mesh imported from 3ds max (fbx format) into unity, i'm texturing it with the creation of the uv map with a script (for each vertex, i give the uv coordinate from the texture)

But as you know, theorically the texture mapping is done by faces (for each face the texture is in a defined place in the texture picture), but in unity this is done by vertex (uv array), so i'm confused to find a solution for the shared vertex between multiples triangles faces (it's the case for all my vertices-they're shared between more than 4 faces-)!

The problem is a flagrant when I affect to vertices form two triangles (sharing a same vertex), uv coordinates of two texture region that are not (one beside the other), so one face is textured with the pixels that exists between the two regions of the image (texture)

Thanks in advance for your help.