Final answer:
The rare form of cancer described by swelling, warmth, or erythema and skin thickening on mammograms is inflammatory carcinoma, which is a type of carcinoma originating from epithelial cells.
Step-by-step explanation:
A rare form of cancer that presents with swelling, warmth, or erythema and mammographically with skin thickening is inflammatory carcinoma. This type of cancer is aggressive and rapidly growing, and symptoms may resemble an infection. A biopsy can provide a definitive diagnosis of cancer, and it is classified based on the type of tissue where the cancer began. Carcinomas are cancers derived from epithelial cells and they account for many types of organ cancers, including breast cancer.