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Let:

##\displaystyle f=\int_{V'} \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'##

where ##V'## is a finite volume in space

##\mathbf{r}=(x,y,z)## are coordinates of all space

##\mathbf{r'}=(x',y',z')## are coordinates of ##V'##

##|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}##

How to prove that:

##\lim\limits_{\Delta x \to 0} \dfrac{f(x+\Delta x,y,z)-f(x,y,z)}{\Delta x}## exist

##\displaystyle f=\int_{V'} \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'##

where ##V'## is a finite volume in space

##\mathbf{r}=(x,y,z)## are coordinates of all space

##\mathbf{r'}=(x',y',z')## are coordinates of ##V'##

##|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}##

How to prove that:

##\lim\limits_{\Delta x \to 0} \dfrac{f(x+\Delta x,y,z)-f(x,y,z)}{\Delta x}## exist