If the cat has a pedigree, it is purebred, otherwise not! A cat without a pedigree, even if born to parents with pedigrees, cannot and is not considered to be purebred by any feline association or any serious breeder.
Pedigree is also important for other reasons: “knowing how to read” a cat's pedigree is a skill that breeders know very well, because only through this document is it possible to study the bloodlines that are behind our cats, to study them strengths and weaknesses and plan their own breed selection activities.
Buy without pedigree?
NEVER buy a purebred cat without a pedigree. You are simply buying a house cat!