In the book of Mathematical Analysis II by Zorich, at page 136, it is asked that
Let $f : I_{a,b} → R$ be a continuous function defined on an interval $I_{a,b} = \{x ∈ \mathbb{R}^n | a_i ≤ x_i ≤ b_i , i = 1, . . . , n\}$, and let $F : I_{a,b} → \mathbb{R}^1 $ be defined by the equality
$$F(x) = \int_{I_{a,x}} f(t) dt,$$
where $I_{a,x} ⊂ I_{a,b}$. Find the partial derivatives of this function with respect to the variables $x_1, . . . , x_n$.
My work:
Directly from the definition of partial derivative,
$$\frac{\partial F}{\partial x_1 } (x) = \lim_{h \to 0} \frac{ \int_{I_{a, x'} } f(t)dt - \int_{I_{a,x}} f(t) dt}{ |h| },$$
where $x' = (x_1 + h, x_2, ..., x_n)$.
$$ = \lim_{h \to 0} \frac{ \int_{I_{x,x'}} f(t)dt + \int_{I_{a, x} } f(t)dt - \int_{I_{a,x}} f(t) dt}{ |h| } \\ = \lim_{h \to 0} \frac{ \int_{I_{x,x'}} f(t)dt }{ |h| },$$
but $\int_{I_{x,x'}} f(t) dt = 0$ even before $h \to 0$, so the limit has to be zero, but this result is wrong clearly, so where is my mistake ?
The mistake lies in the assumption that$$\int_{I_{a,x'}}f(t)\,\mathrm dt=\int_{I_{x,x'}}f(t)\,\mathrm dt+\int_{I_{a,x}}f(t)\,\mathrm dt.$$This would be like asserting that the area of the rectangle whose vertices are $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$ is the sum of the area of the rectangle whose vertices are $(0,0)$, $(1,0)$, $(1,1)$, and $(0,1)$ with the rectangle whose vertices are $(1,1)$, $(2,1)$, $(2,2)$, and $(1,2)$. (Yes, I know that they are squares.)