Peano kernel of a finite quadrature functional

Let $a<b$, let $x_1,\dots ,x_s\in [a,b]$ be a finite indexed list (repetitions are allowed), and let $w_1,\dots ,w_s\in \mathbb{R}$. For every Riemann-integrable $f:[a,b]\to \mathbb{R}$, define the quadrature value and residual by $Q(f):={\sum }_{i=1}^s{w}_{i}f(x_i),L(f):={\int }_a^{b}f(x)dx-Q(f)\mathrm{.}$ Every formal real polynomial's evaluation function is continuous by Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function and integrable by A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion. In the following exactness predicate, $L(p)$ means $L$ applied to that evaluation function. For a natural $r\ge 1$, say that $Q$ is exact through degree $r-1$ if $L(p)=0$ for the zero polynomial and every nonzero formal real polynomial $p$ with degree at most $r-1$ (Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials). Here $r-1$ is the natural predecessor specified by The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$. For any natural $q$, the factorial $q!$ in a real-valued formula denotes its canonical real image under The canonical natural $\iota (n)=n\cdot 1_F$ of a field. This denominator is nonzero: $q!\mathrm{\ne }0$ by The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$, every nonzero natural is a successor by Every nonzero natural number is a successor and therefore is at least $1$ in the natural order Order on the natural numbers, and its canonical real image is positive by Canonical naturals are positive and strictly increasing, hence invertible by The reals form a field. For $t\in [a,b]$ define $g_{r,t}(x):=\{\begin{array}{ll}{\displaystyle \frac{(x-t{)}^{r-1}}{(r-1)!}},& x>t,\\ 0,& x\le t,\end{array}{K}_r(t):=L(g_{r,t})\mathrm{.}$ The power is the natural power of Integer powers $a^m$. For $r\ge 2$, $g_{r,t}$ is polynomial and continuous on each side of $t$ by Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function; the polynomial branch extends to $t$ with value $0$ because $r-1\ge 1$, agreeing with the zero branch. Thus it is continuous on $[a,b]$ and Riemann-integrable by A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion. For $r=1$ it is the indicator of $(t,b]$; if $t=b$ it is zero, and if $t<b$, for each $\epsilon >0$ choose $s\in (t,b)$ with $s-t<\epsilon$ and use the partition with breakpoints $a,t,s,b$ after deleting repeated endpoints. The indicator is constant on all panels except $[t,s]$, where its oscillation is $1$, so its upper-minus-lower sum is at most $s-t<\epsilon$; Riemann's criterion: a bounded $f$ on $[a,b]$ is Darboux integrable if and only if for every real $\epsilon >0$ there is a partition $P$ with $U(f,P)-L(f,P)<\epsilon$ makes it integrable. At $r=1$ the denominator is $0!=1$. Once $K_1$ is shown integrable, changing its values at the finitely many quadrature nodes does not change its integral by Changing an integrable function at finitely many points changes neither its integrability nor its integral.