13 pages, 2 figuresLet 0<λ<1 and I1​=[a0​,a1​),…,Ik​=[ak−1​,ak​) be a partition of the interval I=[0,1) into k≥1 subintervals. Let f:I→I be a map where each restriction f∣Ii​​ is an increasing λ-Lipschitz function for i=1,…,k. We prove that any piecewise increasing contraction f admits at most k periodic orbits, where the upper bound is sharp. Our second result concerns piecewise λ-affine maps. Let b1​,…,bk​ be real numbers. Let Fλ​:I→R be a family of piecewise λ-affine functions, where each restriction Fλ​∣Ii​​(x)=λx+bi​. Under a generic assumption on the parameters a1​,…,ak−1​,b1​,…,bk​ which define Fλ​, we prove that, up to a zero Hausdorff dimension set of slopes 0<λ<1, the ω-limit set of the piecewise λ-affine map fλ​:x∈I→Fλ​(x)(mod1) at every point equals a periodic orbit and there exist at most k periodic orbits