This question is related to my other question boundary term in the integration by parts formula for difference quotient.Now, let us consider two functions $f,g\in C_{0}^{\infty}(\mathbb{R})$ and a time interval $(0,T)$. Let us consider the integration by parts of difference quotients:\begin{align}\int_{0}^{T}\frac{f(x)-f(x-h)}{h}g(x)dx=-\int_{0}^{T}f(x)\frac{g(x+h)-g(x)}{h}dx.\end{align}But since these two are differentable functions, so the difference quotients should converge to the derivative. Hence, passing to the limit implies\begin{align}\int_{0}^{T}f^{\prime}(x)g(x)dx=-\int_{0}^{T}f(x)g^{\prime}(x)dx.\end{align}I do not think this is usually true, because the boundary terms are missing. So I really wonder where does the mistake occur?
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