LAUSR

AbstractLet $${(G, +)}$$(G,+) be an Abelian topological group, D be a subset of G and let a function $${\alpha: D - D \rightarrow \mathbb{R}}$$α:D-D→R be locally bounded above at zero. A function $${f: D \rightarrow \mathbb{R}}$$f:D→R we call $${\alpha}$$α-convex if $$f(z) \leq \frac{f(x)+f(y)}{2}+\alpha(x-y)$$f(z)≤f(x)+f(y)2+α(x-y)for all $${x,y,z \in D}$$x,y,z∈D such that $${x+y=2z}$$x+y=2z. We prove that if $${\alpha(0)=0}$$α(0)=0, $${\alpha}$$α is continuous at zero, D is open and connected, f is $${\alpha}$$α-convex and locally bounded above at a point then f is locally uniformly continuous. We show that the same is true if we replace the assumption that f is locally bounded above at a point by assumption that f is Haar measurable or Baire measurable. We give also Ostrowski-type and Mehdi-type theorem for such functions.