Let f:A→R be a function. Let A = A1 U A2 U...U An is a finite decomposition of the domain A of f such that for some δ0>0 and i≠j in {1,2,...n}, we have |x-y| >= δ0 for all x∈Ai and y∈Aj. Show that if f is uniformly continuous on each of the sets Ai independently, then it is uniformly continuous on A as well.
first we want to consider since f is uniformly continuous on Ai for every i=1,2,...n, then we could get for all ϵ>0, there exist δ>0, for all x,y∈Ai, |x-y|<δ implies |f(x)-f(y)|< ϵ. However, the above information is |x-y| >= δ0, how could I use this in my proof. Can I set δ=min{δ1,δ2,...δn}