A sufficient condition $\int \frac{df}{dt}dx = \frac{d}{dt} \int fdx$ holds is Will moving differentiation from inside, to outside an integral, change the result?,
or When do the partial derivative and integral of different variables commute?.
On the other hand, an example of $\int \frac{df}{dt}dx \neq \frac{d}{dt} \int fdx$ is Derivative commuting over integral.
This example, $f$ is discontinuous function.
I want to know example of continuous function $f$ such that
- $x \mapsto f(t, x)$ is integrable for all $t$,
- $t \mapsto f(t, x)$ is differentiable for all $x$,
- $\int \frac{df}{dt}dx \ $ and $\ \frac{d}{dt} \int fdx$ exist in $\mathbb{R}$, and
- $\int \frac{df}{dt}dx \neq \frac{d}{dt} \int fdx$.
i.e. an integrable function $g$ such that $$\left| \frac{\partial f}{\partial t}(t, x) \right| \le g(x).$$
doesn't exist. (Will moving differentiation from inside, to outside an integral, change the result?)
Any advise would be appreciated.
