We are told that if we assume $$E(u|x) = 0$$, then the population regression function can be interpreted as $E(y|x) = a + bx$, where "$a$" and "$b$" are the population parameters.
However I can construct a counterexample where this doesn't hold:
$$\begin{array}{c|c|c|} 
 & \text{x} & \text{y} \\ \hline
\text{} & 1 & 12 \\ \hline
\text{} & 2 & 14 \\ \hline
\text{} & 3 & 16\\ \hline
\text{} & 4 & 20\\ \hline
\text{} & 5 & 25\\ \hline
\text{} & 6 & 29\\ \hline
\text{} & 7 & 31\\ \hline
\text{} & 8 & 40\\ \hline
\text{} & 9 & 20\\ \hline
\end{array}$$

Suppose the above is a population dataset (I'm assuming I know exactly the population, there is no sample). $E(y|x) = y$, for instance $E(y|x=1) = 12$ since $x=1$ has only 1 $y$ value which is equal to 12. If this is true whats the population regression line?

A linear regression yields gives us $Y = 10.87 + 2.41x$. However, this does not satisfy the property of $E(u|x)=0$, clearly for $x=1$, the predicted value is not equal to 12, there is an error. So how is it that linear regression satisfies $E(u|x)$?

Does this mean $E(u|x) = 0$ doesn't hold in the population and is just an assumption? 

Even under the assumption that $E(u|x)$ is true, with the population dataset above I cannot find a linear equation that satisfies this. What are the implications of this in real world examples when it doesn't seem to hold?