A function \( f : \mathbb{R} \to \mathbb{R} \) satisfies
\[
f\!\left(\frac{x+y}{3}\right) = \frac{f(x) + f(y) + f(0)}{3} \quad \text{for all } x,y \in \mathbb{R}.
\]
If \( f'' \) is differentiable at \( x = 0 \), then \( f \) is:
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Mean-type functional equations:
Symmetric averaging usually implies linearity.
Try polynomial substitution and compare degrees.