Proposition 2

Let I be an interval of $\mathbf{R}$ and $f$ a mapping from $X\times I$ into E such that

(i) For every $t\in I$, the mapping $x\mapsto f(x, t)$ of X into E is $\nu$-integrable;

(ii) For every $x\in X$, the mapping $t\mapsto f(x, t)$ of I into E admits a derivative, denoted by $t\mapsto f'(x, t)$;

(iii) There exists a positive $\nu$-integrable function $g$ on X such that $\|f'(x, t)\|\leqslant g(x)$ for every $(x, t)\in X\times I$.

Then the mapping F of I into E defined by

is differentiable in I, and for every $t\in I$, one has

Let $t_0\in I$ and let J be an interval of $\mathbf{R}$ contained in I which is a neighbourhood of $t_0$ in I. Let $h: X\times J\rightarrow E$ be the mapping defined by $(x, t)\mapsto (f(x, t)-f(x, t_0))/(t-t_0)$ for $(x, t)\in X\times (J-\{t_0\})$ and $(x, t_0)\mapsto f'(x, t_0)$ for $x\in X$. Let $x\in X$. One has $\|h(x, t_0)\|\leqslant g(x)$ and, for every $t\not =t_0$, one has $\|h(x, t)\|\leqslant g(x)$ by FVR, I, p. 23, th. 2. By definition of the derivative, the proposition follows from corollary 1 of INT, IV, p. 144, § 4, n$^o3$, applied to the mapping $h$.

Let us resume the notation of VAR, R1, 2.4, p. 19 concerning partial derivatives.