Differentiation of improper integrals defined on the whole real line. I am considering improper Riemann integrals (not Lebesgue integrals, mind you) of the form $$\int_{-\infty}^\infty f(t,x)dt,$$
with $f:\mathbf{R}\times\Omega\rightarrow\mathbf{R}$ continuous ($\Omega$ an open set in $\mathbf{R}$). What are sufficent conditions on $f$ to justify 
$$\frac{d}{dx}\int_{-\infty}^\infty f(t,x)dt=\int_{-\infty}^\infty \frac{\partial}{\partial x}f(t,x)dt?$$
References are welcome. It seems to me that no book includes this :(
 A: A sufficient condition is that the integral $\int_{-\infty}^\infty \frac{\partial}{\partial x}f(t,x)\,dt$ is  uniformly convergent with respect to parameter $x$ (in some neighborhood of the point $x$ that you are interested in). This means you can bound the tail of integral by $\epsilon$ using the same size of tail for all $x$. 
Googling "uniformly convergent" and "improper integral" brings up proofs of the result, such as Theorem 5 here.
A: Hint: In short, at a point $y_0$, we have:
$$\frac{d}{dy} \int_a^b f(x,y)dx =\int_a^b  \frac{\partial}{\partial y} f(x,y)dx,$$
if $f(x,y)$ and $\frac{\partial}{\partial y} f(x,y)$ are continuous of both variables ($x$ in its integration range and $y$ in an interval around $y_0$), and are, respectively, bounded in absolute value by two functions $g(x)$ and $h(x)$, both independent of $y$, such that $\int_a^b g(x)dx$ and $\int_a^b h(x)dx$ converge.
See this paper for reference (Theorem 10.3) and for many examples of its usage, and Serge Lang's Undergraduate Analysis, Chapter XIII - Improper Integrals, Section 3 - Interchanging Derivatives and Integrals.
