Differentiation wrt integral boundaries

[nickstern]

Hi:Let $f,g$ and $h$ be continuous real valued function with domain $X \times Y$ where $X$ and $Y$ are compact sets.
Let me define the set
$S(x) = \{ y : h(y,x) \geq 0 \} $
and
$\bar{S}(x) = \{ y : h(y,x) \leq 0 \} $

Also for any $(x,y) \in X \times Y $ such that $h(y,x)=0$, it holds $f(y,x)=g(y,x)$

Let the function $m$ be defined as:
\begin{equation*}
m(x) = \int_{y \in S(x)} f(y,x) dy + \int_{y \in \bar{S}(x)} g(y,x) dy
\end{equation*}I can differentiate the term inside the integral if $f$ and $g$ and their partial derivatives are continuous in $x$ and $y$

What I do not know is the assumptions required on $h$ to have $m$ differentiable.
My intuition is that since at every point where $h(y,x)=0, f(y,x)=g(y,x)$ differentiation with respect to the limits of the integral is not a problem but I want to make the argument formal.

Any help would be much appreciated! Thanks!

Nick

 

[jvicens]

Dear Nick,

Thank you for your post. Your intuition is correct - if $h$ is continuous and $f$ and $g$ are continuous and have continuous partial derivatives, then $m$ will be differentiable. This can be seen by applying the Leibniz integral rule, which states that if a function $F(x,y)$ is continuous and has continuous partial derivatives, then $\frac{d}{dx}\int_{a(x)}^{b(x)}F(x,y)dy = \int_{a(x)}^{b(x)}\frac{\partial}{\partial x}F(x,y)dy + F(b(x),x)\frac{db}{dx}-F(a(x),x)\frac{da}{dx}$.

In your case, $F(x,y) = f(y,x)$ when $h(y,x) \geq 0$ and $F(x,y) = g(y,x)$ when $h(y,x) \leq 0$. Since $f$ and $g$ have continuous partial derivatives, $\frac{\partial}{\partial x}F(x,y)$ will also be continuous and thus $m(x)$ will be differentiable.

However, if $h$ is not continuous, then the Leibniz integral rule may not apply and it may not be possible to guarantee the differentiability of $m(x)$. In this case, additional assumptions on the behavior of $h$ may be necessary.

I hope this helps. Let me know if you have any further questions.