MathDB

Let be a twice-differentiable function

f:\mathbb{R}\longrightarrow\mathbb{R}

that has the properties that:

\text{(i) supp} f''=f\left(\mathbb{R}\right)

\text{(ii)}\exists g:\mathbb{R}\longrightarrow\mathbb{R} \forall x\in\mathbb{R} f(x+1)=f(x)+f'\left( g(x)\right)\text{ and } f'(x+1)=f'(x)+f''\left( g(x)\right)
Prove that: a) any such

g

is injective. b)

f

is of class

C^{\infty } ,

and for any natural number

n,

any real number

x

and any such

g,

f^{(n)}(x+1)=f^{(n)}(x)+f^{(n+1)}\left( g(x)\right) .

Laurențiu Panaitopol

Problem Statement