Suppose function $f:[0,1]\rightarrow \mathbb{R}$ is approximated as shown in the figure.
$$|f\rangle \rightarrow \begin{pmatrix} f_1\\ f_2\\ \vdots\\ f_N \end{pmatrix}$$ The derivative defined as $$\frac{df}{dx}=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ In this case, $$\left.\frac{df}{dx}\right|_{i^\text{th} \text{bin}}=\lim_{N\rightarrow \infty}\frac{f_{i+1}-f_i}{1/N}=\lim_{N\rightarrow \infty}N(f_{i+1}-f_i)$$
So I can write the derivative operator as a matrix like
$$D=N\begin{pmatrix} -1 & 1 & 0& \cdots \\ 0 & -1 & 1 & \cdots \\ \vdots&0&-1&\cdots \end{pmatrix}_{(N-1)\times (N-1)}$$
Now it is said, in the problem that I'm working on, that as $N\rightarrow \infty$ the operator $D$ becomes an operator $\mathcal{D}$ with the property $$\int \mathcal{D}(x,x')f(x')dx'=\frac{df}{dx}$$
I don't understand, How does one show this? Can anyone help me with this?