Let \(\theta\) be a fixed constant. Characterize all functions \(f:\mathcal R\to \mathcal R\) such that \(f”(x)\) exists for all real \(x\) and for all real \(x,y\), \[ f(y)=f(x)+(y-x)f'(x)+ \frac{(y-x)^2}{2} f”(\theta y + (1-\theta) x).\]
Find all continuous functions \(f : \mathbb{R} \to \mathbb{R}\) satisfying
\[
f(x) = f(x^2 + \frac{x}{3} + \frac{1}{9} )
\]
for all \( x \in \mathbb{R} \).
Let \( \mathbb{Z}^+ \) be the set of positive integers. Suppose that \( f : \mathbb{Z}^+ \to \mathbb{Z}^+ \) satisfies the following conditions.
i) \( f(f(x)) = 5x \).
ii) If \( m \geq n \), then \( f(m) \geq f(n) \).
iii) \( f(1) \neq 2 \).
Find \( f(256) \).