Putnam 2010, Problem A-2

Find all differentiable functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[
f'(x) = \frac{f(x+n) - f(x)}{n}
\]
for all real numbers \( x \) and all positive integers \( n \).

Solution:
The only such functions are those of the form $f(x) = cx + d$ for some real numbers $c, d$ (for which the property is obviously satisfied). To see this, suppose that $f$ has the desired property. Then for any $x \in \mathbb{R}$,
$$2f'(x) = f(x+2) - f(x)$$
$$= (f(x+2) - f(x+1)) + (f(x+1) - f(x))$$
$$= f'(x+1) + f'(x).$$

Consequently, $f'(x+1) = f'(x)$.

Define the function $g : \mathbb{R} \to \mathbb{R}$ by $g(x) = f(x+1) - f(x)$, and put $c = g(0)$, $d = f(0)$. For all $x \in \mathbb{R}$, $g'(x) = f'(x+1) - f'(x) = 0$, so $g(x) = c$ identically, and $f'(x) = f(x+1) - f(x) = g(x) = c$, so $f(x) = cx + d$ identically as desired.