TITLE: Derivative commuting over integral
QUESTION [9 upvotes]: Can a derivative operation commute over an integral operation irrespective of the properties of the function under the integral ?

REPLY [11 votes]: Not in general.  I recommend Gelbaum and Olmsted's Counterexamples in Analysis, which is where I turned to find a counterexample to your question.  Namely, example 15 on page 123 is titled

A function $f$ for which $d/dx\int_a^b f(x,y)dy\neq\int_a^b[\partial/\partial x f(x,y)]dy$, although each integral is proper.

The example is
$$f(x,y) = \left\{
     \begin{array}{lr}
       \frac{x^3}{y^2}e^{-x^2/y} & : y>0, \\
       0 & : y=0, 
     \end{array}
   \right.
$$
integrated with respect to $y$ from $0$ to $1$.  Actually, differentiating under the integral sign works here except where $x=0$.
The function and its partial derivative are not jointly continuous.  When they are jointly continuous, differentiation and integration commute.