This question was asked on Cross Validated where it received little attention and no comments or answers, but as it is purely mathematically oriented it may well be more suitable here.
Consider a linear regression (based on least squares) on two predictors including an interaction term: $$Y=(b_0+b_1X_1)+(b_2+b_3X_1)X_2$$
$b_2$ here corresponds to the conditional effect of $X_2$ when $X_1=0$. A common mistake is to understand $b_2$ as being the main effect of $X_2$, i.e. the average effect of $X_2$ over all possible values of $X_1$.
Now let's assume that $X_1$ was centered, that is $\overline{X_1}=0$. It becomes now true that $b_2$ is the average effect of $X_2$ over all possible values of $X_1$, in the sense that $\overline{b_2+b_3X_1}=b_2$. In such conditions, the meaning given to $b_2$ is nearly indistinguishable from the meaning that we would give to the effect of $X_2$ in a simple regression (where $X_2$ would be the only factor, let's call this effect $B_2$).
In practice, it seems that $b_2$ and $B_2$ are reasonably close to each other.
Are there any "common knowledge" examples of situations where $B_2$ and $b_2$ are remarkably far from each other?
Are there any known upper bounds to $|b_2-B_2|$?