Firstly I think you mean continuous and discontinuous variation!
Yes, they can be (not always) transmitted to the next generation - if the genome of an organism is altered, this will be reflected in the gametes (as part of a haploid number of chromosomes, given), and therefore might pass to the offspring, depending on which alleles come from the affected parent and which from the other parent.
An example is the transmission of antibiotic resistance that has been acquired by a few bacteria through random mutation, which is transmitted both VERTICALLY and horizontally to the whole population of bacteria over time.
Whether any variation is continuous or discontinuous depends on the phenotype being looked at. A discrete feature like brown/white moths in the classic example, that is determined by a single gene, will lead to discontinuous variation, while some non-discrete feature like height or weight (where the phenotype is not "black or white") will undergo continuous variation.