Theorem A bounded function f:[a,b]→Rf:[a, b] \rightarrow Rf:[a,b]→R is Riemann integrable iff for every ϵ>0\epsilon > 0ϵ>0 there exists a partition PϵP_\epsilonPϵ of [a,b][a,b][a,b], such that: U(f,Pϵ)−L(f,Pϵ)≤ϵU(f, P_\epsilon) - L(f, P_\epsilon) \le \epsilonU(f,Pϵ)−L(f,Pϵ)≤ϵ