Let $T:X\to{\Bbb R}$ be a piecewisemonotonic map, where $X$ is a finite union of closedintervals. Define $R(T)=\bigcap_{n=0}^{\infty}\overline{T^{-n}X}$, and suppose that $(R(T),T)$ hasa unique maximal measure $\mu$. The influence ofsmall perturbations of $T$ on the maximal measure isinvestigated. If $(R(T),T)$ has positive topologicalentropy, and if a certain stability condition issatisfied, then every piecewise monotonic map$\tilde{T}$, which is contained in a sufficientlysmall neighbourhood of $T$, has a unique maximalmeasure $\tilde{\mu}$, and the map$\tilde{T}\mapsto\tilde{\mu}$ is continuousat $T$.