real analysis – A new equivalent characterization of Riemann-Integrability

Question:

Given two bounded functions $ , f:[a,b] to mathbb R ; $ and $ ; theta: (0, b-a]at [0,1]$.

assume $ , P: a = x_0 <x_1 <⋯ <x_n = b ; $ is a score of $[a,b]$.

Let $ , Δx_k = x_k – x_ {k – 1} $ and $ ; ‖P‖ = max_ {1≤k≤n} , Δx_k $.

Yes $$ exists A in mathbb R, , forall epsilon> 0, , exists delta> 0: , forall P ; bigl (P‖ < delta bigl) ; Longrightarrow , Biggl green { sum_ {k = 1} ^ nf bigl (x_ {k – 1} + + theta (Δx_k) Δx_k bigl) Δx_k-A} Biggl green < epsilon , , $$

that is to say. $$ sum_ {k = 1} ^ n f bigl (x_ {k – 1} + theta (Δx_k) Δx_k bigl) Δx_k to A ; bigl (P‖ to 0 bigl) ,, $$

so can we still conclude that $ , f ; $ Is Riemann integrable?

Context:

I note that Kristensen, Poulsen, and Reich (A Characterization of Riemann-Integrability, The American Mathematical Monthly, 69, No. 6, pp. 498-505) gave a similar result with the Darboux property, but my question does not require to satisfy the property of Darboux.

Can any one help me? Any tip or solution would be much appreciated. Thanks in advance!