Question:
Given two bounded functions $\,f:[a,b]\to \mathbb R\;$ and $\;\theta:(0,b-a]\to [0,1]$.
Suppose $\,P:a=x_0<x_1<⋯<x_n=b\;$ is a partition of $[a,b]$.
Let $\,Δx_k=x_k−x_{k−1}$ and $\;‖P‖=\max_{1≤k≤n}\,Δx_k$.
If $$\exists A\in\mathbb R,\,\forall\epsilon>0,\,\exists\delta>0:\,\forall P\;\bigl(‖P‖<\delta\bigl)\;\Longrightarrow \,\Biggl\vert{\sum_{k=1}^n f\bigl(x_{k−1}+\theta(Δx_k)Δx_k\bigl)Δx_k-A}\Biggl\vert<\epsilon\,,$$
i.e. $$\sum_{k=1}^n f\bigl(x_{k−1}+\theta(Δx_k)Δx_k\bigl)Δx_k\to A\; \bigl(‖P‖\to 0\bigl)\,,$$
then can we always conclude that $\,f\;$ is Riemann-integrable?
Background:
I notice that Kristensen, Poulsen and Reich (A characterization of Riemann-Integrability, The American Mathematical Monthly, vol.69, No.6, pp. 498-505) have proved a similar result with the Darboux property, but my question need not satisfy Darboux property.
Can anybody help me? Any hints or solution would be much appreciated. Thanks in advance!