Can a derivative operation commute over an integral operation irrespective of the properties of the function under the integral ?
Derivative commuting over integral
8
$\begingroup$
calculus
integration
-
2The answer is no. You could read here: http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign – 2010-11-03
1 Answers
11
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.
-
0Not in general. If you didn't assume separate continuity, for example, you could write down a function that is everywhere discontinuous but constant in $x$ for each fixed $y$, so that the partial derivative is everywhere $0$. If you mean in addition to assuming continuity in $y$ for each fixed $x$, then I don't know a counterexample off hand, but suspect it is not always true (correct me if I'm wrong). – 2010-11-03