Is the following derivative defined? I am new to this site so I am not sure if this is the right place to be asking this question but I will try anyway. I am reading an economics paper for my research and the author does the following:
$$\frac{\partial}{\partial C_t(j)} \int_0^1 P_t(j) C_t(j) dj = P_t(j)$$
I feel that this derivative is not properly defined but I am probably missing something obvious because the author knew what he was doing. Could someone please tell me if this is a legitimate derivative?
Thanks,

That derivative can be properly defined if and only if

there exists an appropriate pair of values $t$ and $j$ such that $\: C_t(j) = 0 \:$
and
all appropriate values of $t$ such that

[there exists an appropriate value of $j$ such that $\: C_t(j) = 0 \:$]

give the same value of the integral
and
$0$ is a limit point of the set of values taken by $C_t(j)$ for appropriate values of $t$ and $j$

("appropriate values" are those associated with the definition of $C_t(j)$, and even if the derivative "can be properly defined", it may still fail to exist, as is the case with non-differentiable functions).

If either of the first two fail there is not a unique "base value" to plug into the difference quotient.

If the last one fails, it is not clear what the definition of "limit" would be.
