A continuous extrusion foaming process comprising the following steps: (a) providing a polymer melt in an extruder of an extrusion foaming line, the polymer melt comprising a polymer composition that has a softening temperature and that consists of all of the polymers in the polymer melt; (b) introducing carbon dioxide into the polymer melt within the extrusion foaming line at an initial addition pressure while mixing the polymer melt and carbon dioxide together and while the polymer melt is at an initial addition temperature that is above the softening temperature of the polymer composition to form a polymer/carbon dioxide mixture wherein the total amount of carbon dioxide added to the polymer melt exceeds the amount of carbon dioxide that is soluble in the polymer composition at the initial addition temperature and initial addition pressure and the carbon dioxide is dispersed throughout the polymer composition; (c) cooling the polymer/carbon dioxide mixture to a dissolving temperature that is below the initial addition temperature while keeping the pressure around the polymer/carbon dioxide mixture between the initial addition pressure and a dissolving pressure that is equal to or below the initial addition pressure, wherein all of the carbon dioxide in the polymer/carbon dioxide mixture is soluble in the polymer composition at the dissolving temperature and dissolving pressure; and (d) extruding the polymer/carbon dioxide mixture through an extrusion die into an expansion region having an expansion pressure that is lower than the dissolution pressure such that the polymer/carbon dioxide mixture experiences a pressure drop of at least five MegaPascals at a rate of at least ten MegaPascals per second as it exits the extrusion die and expands into a polymeric foam having an average transverse cell size that is less than one micrometer; wherein the process is free from adding solid carbon dioxide to the polymer and wherein the polymer/carbon dioxide mixture does not experience a pressure greater than the initial addition pressure during the extrusion foaming process, and wherein carbon dioxide is added to the polymer composition at more than one location within the extrusion foaming line and wherein carbon dioxide is first added to the polymer melt at an initial addition pressure and subsequent additions of carbon dioxide to the polymer composition occur at a pressure at or below the initial addition pressure and at or above the dissolving pressure, and further wherein the temperature of the polymer composition at the first addition location of carbon dioxide is at the initial addition temperature and below the initial addition temperature during subsequent additions of carbon dioxide. (a) providing a polymer melt in an extruder of an extrusion foaming line, the polymer melt comprising a polymer composition that has a softening temperature and that consists of all of the polymers in the polymer melt; (b) introducing carbon dioxide into the polymer melt within the extrusion foaming line at an initial addition pressure while mixing the polymer melt and carbon dioxide together and while the polymer melt is at an initial addition temperature that is above the softening temperature of the polymer composition to form a polymer/carbon dioxide mixture wherein the total amount of carbon dioxide added to the polymer melt exceeds the amount of carbon dioxide that is soluble in the polymer composition at the initial addition temperature and initial addition pressure and the carbon dioxide is dispersed throughout the polymer composition; (c) cooling the polymer/carbon dioxide mixture to a dissolving temperature that is below the initial addition temperature while keeping the pressure around the polymer/carbon dioxide mixture between the initial addition pressure and a dissolving pressure that is equal to or below the initial addition pressure, wherein all of the carbon dioxide in the polymer/carbon dioxide mixture is soluble in the polymer composition at the dissolving temperature and dissolving pressure; and (d) extruding the polymer/carbon dioxide mixture through an extrusion die into an expansion region having an expansion pressure that is lower than the dissolution pressure such that the polymer/carbon dioxide mixture experiences a pressure drop of at least five MegaPascals at a rate of at least ten MegaPascals per second as it exits the extrusion die and expands into a polymeric foam having an average transverse cell size that is less than one micrometer;