Suppose that we regress a variable $Y$ on two independent variables $X_1$ and $X_2$. Since we have included $X_2$ in the regression, the obtained $\hat{\beta_1}$ coefficient is often said to be the effect of $X_1$ on $Y$ ‘holding $X_2$ fixed’ or 'controlling for $X_2$'. This is then said to eliminate any omitted variable bias that is due to $X_2$.
To investigate this more closely, observe that we can hold $X_2$ fixed at a variety of different values. Restrict attention to those values that reappear at least once across the dataset (e.g. because there are two datapoints for which $X_{2i} = 5$). In such cases, we can conduct a simple regression of $Y$ on $X_1$ solely using those datapoints for which (say) $X_{2i} = 5$. We can then repeat this regression for all possible $X_2$ values (assume they reappear at least once). It is natural to conjecture that our obtained $\beta_2$ estimate should be similar (identical?) to the average of the $\beta_2$ coefficients obtained from our simple regressions. Is this indeed the case?