- #1
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Well here is my (I hope successful this time ) attempt at 10. e)
We consider the function ##f(x,y)=\begin{cases}\dfrac{e^{-x|y|}}{y}, y\neq 0 \\ 0, y=0\end{cases}##.
For this function for ##y\neq 0## it is for any ##x##, ##\frac{\partial f}{\partial x}=-\frac{|y|}{y}e^{-x|y|}##.
Also it should be clear that due to the infinite discontinuity at the points ##(x,0)## the integral ##\int_\mathbb{R}f(x,y)dy## does not converge hence the left hand side of the condition of 10. e) is infinite (or undefined).
The right hand side of the condition is ##\int_\mathbb{R}-\frac{|y|}{y}e^{-x|y|}dy=\int_{-\infty}^0e^{xy}dy+\int_0^{+\infty} -e^{-xy} dy=0## ,
So the condition is true for this ##f##.
We consider the function ##f(x,y)=\begin{cases}\dfrac{e^{-x|y|}}{y}, y\neq 0 \\ 0, y=0\end{cases}##.
For this function for ##y\neq 0## it is for any ##x##, ##\frac{\partial f}{\partial x}=-\frac{|y|}{y}e^{-x|y|}##.
Also it should be clear that due to the infinite discontinuity at the points ##(x,0)## the integral ##\int_\mathbb{R}f(x,y)dy## does not converge hence the left hand side of the condition of 10. e) is infinite (or undefined).
The right hand side of the condition is ##\int_\mathbb{R}-\frac{|y|}{y}e^{-x|y|}dy=\int_{-\infty}^0e^{xy}dy+\int_0^{+\infty} -e^{-xy} dy=0## ,
So the condition is true for this ##f##.
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