when can we interchange integration and differentiation
Let $f$ be a Riemann Integrable function over $\mathbb{R}^2$. When can we do this?
$$\frac{\partial}{\partial\theta}\int_{a}^{b}f(x,\theta)dx=\int_{a}^{b}\frac{\partial}{\partial\theta}f(x,\theta)dx$$
(Here, $a$ and $b$ are not a function of $\theta$.)
In the problem, which I am solving recently, are like this:
$f_{\theta}(x)$, here $\theta$ is constant and $\theta\in\mathbb{R}$ (usually). For example $f_{\theta}(x)=x^2\theta$. So, I am blindly interchanging integration and differentiation because of continuity over $\theta$. But I want to know little bit more.
Also, what happens if $a$ and $b$ are function of $\theta$? Thanks.
$\endgroup$ 21 Answer
$\begingroup$You may interchange integration and differentiation precisely when Leibniz says you may. In your notation, for Riemann integrals: when $f$ and $\frac{\partial f(x,t)}{\partial x}$ are continuous in $x$ and $t$ (both) in an open neighborhood of $\{x\} \times [a,b]$.
There is a similar statement for Lebesgue integrals.
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