Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that
\[f(x+t) - f(x) = P(x)\]
for all $x \in \mathbb{R}$.
chaotic_iak wrote:
Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that
\[f(x+t) - f(x) = P(x)\]
for all $x \in \mathbb{R}$.
Choose any $t\ne 0$ and define $f(x)$ as any function you want over $[0,t)$ and by the induction formulas :
$f(x+t)=f(x)+P(x)$
$f(x-t)=f(x)-P(x)$
And you get all the solutions, built piece per piece.
pco wrote:
Choose any $t\ne 0$ and define $f(x)$ as any function you want over $[0,t)$ and by the induction formulas :
$f(x+t)=f(x)+P(x)$
$f(x-t)=f(x)-P(x)$
And you get all the solutions, built piece per piece.
I think it is just a typo, but it should be:
...
$f(x-t)=f(x)-P(x-t)$.