Exercise 15, § 3

Let $I$ be an open interval in $\mathbf{R}$, and $\mathbf{f}$ a vector function defined and continuous on $I$; suppose that there are $n$ vector functions $g_i$ ($1 \leq i \leq n$) defined on $I$, and such that the function of $x$

tends uniformly to 0 on every compact interval contained in $I$ as $h$ tends to 0

a) We put $\mathbf{f}_p(x,h) = \Delta^p \mathbf{f}(x;h,h,\ldots,h)$ (I, p. 40, exerc. 6). Show that, for $1 \leq p \leq n$, $(1/h^p) \mathbf{f}_p(x,h)$ tends uniformly to $\mathbf{g}_p(x)$ on every compact subinterval of $I$ as $h$ tends to 0, and that the $\mathbf{g}_p$ are continuous on $I$ (prove this successively for $p = n, p = n-1$, etc.)

b) Deduce from this that $\mathbf{f}$ has a continuous $n^{th}$ derivative and that $\mathbf{f}^{(p)} = \mathbf{g}_p$ for $1 \leq p \leq n$ (taking account of the relation $\mathbf{f}_{p+1}(x,h) = \mathbf{f}(x+h,h) - \mathbf{f}_p(x,h)$).