We study the evolution of specialization in a spatially continuous (one-dimensional) environment divided into two habitats; we use a general trade-off function relating fitnesses in the two habitats and illustrate our results with two classical trade-off functions. We show that the population can either reach an intermediate value of the trait and be moderately adapted to both habitats (1 generalist), or split into two locally adapted subpopulations (2 specialists). We recover the qualitative results obtained with simpler metapopulation models with island migration: the evolutionary outcome depends on the concavity of the trade-off, on the proportion of each habitat and on migration. Our quantitative prediction on migration, however, depends on isolation by distance. Our spatially explicit model may thus be particularly useful to describe the evolutionary dynamics of specialization in, perhaps, more realistic ecological scenarios.