The epigenome, including DNA methylation, is stably propagated during mitotic division. However, single-cell clonal expansion produces heterogeneous methylomes, thus raising the question of how the DNA methylome remains stable despite constant epigenetic drift. Here, we report that a clonal population of DNA (cytosine-5)-methyltransferase 1 (DNMT1)-only cells produces a heterogeneous methylome, which is robustly propagated on cell expansion and differentiation. Our data show that DNMT1 has imprecise maintenance activity and possibly possesses weak de novo activity, leading to spontaneous 'epimutations'. However, these epimutations tend to be corrected through a neighbor-guided mechanism, which is likely to be enabled by the environment-sensitive de novo activity ('tuner') and maintenance activity ('stabilizer') of DNMT1. By generating base-resolution maps of de novo and maintenance activities, we find that H3K9me2/3-marked regions show enhanced de novo activity, and CpG islands have both poor maintenance and de novo activities. The imprecise epigenetic machinery coupled with neighbor-guided correction may be a fundamental mechanism underlying robust yet flexible epigenetic inheritance.