A species’ range can be limited when there is no genetic variation for a trait that allows for adaptation to more extreme environments. We study how range expansion occurs by the establishment of a new mutation that affects a quantitative trait in a spatially continuous population. The optimal phenotype for the trait varies linearly in space. The survival probabilities of new mutations affecting the trait are found by simulation. Shallow environmental gradients favour mutations that arise nearer to the range margin and that have smaller phenotypic effects than do steep gradients. Mutations that become established in shallow environmental gradients typically result in proportionally larger range expansions than those that establish in steep gradients. Mutations that become established in populations with high maximum growth rates tend to originate nearer to the range edge and to cause relatively smaller range expansion than mutations that establish in populations with low maximum growth rates. Under plausible parameter values, mutations that allow for range expansion tend to have large phenotypic effects (more than one phenotypic standard deviation) and cause substantial range expansions (15% or more). Sexual reproduction allows for larger range expansions and adaptation to more extreme environments than asexual reproduction.