Let's consider a function $$ f:\mathbb{R}^2\to\mathbb{R} $$ And suppose it is smooth, then for a given $M>0$ and $y\geq 0$, according to Fubini's theorem, we have that $$ \int_{-M}^M\partial_yf(x,y)dx=\frac{d}{dy}\int_0^y\int_{-M}^M\partial_yf(x,y)dxdy=\frac{d}{dy}\int_{-M}^M\int_0^y\partial_yf(x,y)dydx=\frac{d}{dy}\int_{-M}^Mf(x,y)dx $$ Then when $M\to+\infty$ if the limit exists we would have that $$ \int_\mathbb{R}\partial_yf(x,y)dx=\frac{d}{dy}\int_\mathbb{R}f(x,y)dx $$ I would like to know if the reasoning above is correct, any hint would be appreciated.
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How do you know that you can exchange the order of $d/dy$ and $\lim\limits_{M\to\infty}$? – Ningxin Jun 24 '20 at 12:33
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That's what I would like to know if we can do – CaptainNemo Jun 24 '20 at 15:22
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@QiyuWen I think you need the improper integral to converge uniformly – Divide1918 Jun 24 '20 at 17:02
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What does that mean? – CaptainNemo Jun 24 '20 at 21:01