Cyanosis

Is this frostbite that you're talking about?

Cyanosis is caused by low oxygen levels but there is more than one way to have a low oxygen level. The blood has to be properly oxygenated AND it has to be circulated to each cell.

Think in terms of what the body's physiological response to cold is for. It is to minimize loss of body heat. A major heat regulating process is heat loss through the skin. Our capillaries contract to reduce this loss so blood circulation is reduced. The body will protect core heat at the expense of our extremities so at some point the capillaries no longer carry enough blood to these areas (no matter how oxygenated) and the symptoms of cyanosis occurs.

Cyanosis is not limited to the extremities. In cases where it is caused by a blood clot, for instance, the cyanosis can occur wherever the blood flow has been affected.