Rather than requiring endogenous or external mitogens to continue the cell cycle, cancer cells are able to grow, survive, and replicate without mitogens. Cancer cells may lose their dependence on external mitogens by a variety of pathways. 
First, cancer cells can produce their own mitogens, a term called autocrine stimulation. This can result in a deadly positive feedback loop - tumor cells produce their own mitogens, which stimulate more tumor cells to replicate, which can then produce even more mitogens. For example, consider one of the earliest oncogenes to be identified, p28sis from the simian sarcoma virus, which causes tumorigenesis in the host animal. Scientists found that p28sis has a nearly identical amino acid sequence as human platelet-derived growth factor (PDGF). Thus, tumors formed by the simian sarcoma virus are no longer dependent on the fluctuations of PDGF that control cell growth; instead, they can produce their own mitogens in the form of p28sis. With enough p28sis activity, the cells can proliferate without restriction, resulting in cancer.
Second, cancer cells can have mutated cell-surface receptors for mitogens. The protein kinase domain found on mitogenic receptors is often hyperactivated in cancer cells, remaining turned on even in the absence of external mitogens. Additionally, some cancers are associated with an overproduction of mitogenic receptors on the cell surface. With this mutation, cells are stimulated to divide by abnormally low levels of mitogens. One such example is HER2, a receptor tyrosine kinase that responds to the mitogen EGF. Overexpression of HER2 is common in 15-30% of breast cancers, allowing the cell cycle to progress even with extremely low concentrations of EGF. The overexpression of kinase activity in these cells aids in their proliferation. These are known as hormone-dependent breast cancers, as the kinase activation in these cancers is connected to exposure to both growth factors and estradiol.
Third, downstream effectors of mitogenic signaling are often mutated in cancer cells. An important mitogenic signaling pathway in humans is the Ras-Raf-MAPK pathway. Mitogenic signaling normally activates Ras, a GTPase, that then activates the rest of the MAPK pathway, ultimately expressing proteins that stimulate cell cycle progression. It is likely that most, if not all, cancers have some mutation in the Ras-Raf-MAPK pathway, most commonly in Ras. These mutations allow the pathway to be constitutively activated, regardless of the presence of mitogens.