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so, on normal points they are tangent vectors on some point in the chosen direction. how about in critical points, where there shouldnt be any on a geometrical standpoint? can i say they exist if i get them with the definition? or the result i get has no value?

like for example on f(0,0), i get 0 for any derivative. do i say they exist and their value is 0 or what?

when a directional derivative dont exist?