Final answer:
The layer of the cutaneous membrane that produces sweat is the dermis, containing the eccrine sweat glands. The dermis is vascularized and connected to the epidermis by the basement membrane, and it is involved in bleeding when cut.
Step-by-step explanation:
The layer of the cutaneous membrane responsible for producing sweat is the dermis. This layer contains the eccrine sweat glands, which are responsible for regulating body temperature through thermoregulation. Unlike the epidermis, which is the outermost layer and lacks blood vessels, or the hypodermis, which primarily contains fat and connective tissue, the dermis is rich in both blood vessels and sweat glands.
The basement membrane lies between the dermis and epidermis, making it crucial in connecting these two layers. If you cut yourself and notice bleeding, you have cut into at least the papillary dermis, which is part of the overall dermis layer and contains blood vessels.