This question is related to another question of mine Invariance of strategy-proof social choice function when peaks are made close from solution, and it revolves around the use of limit arguments with non-continuous function.
The question is drawn from an argument in the proof of lemma 2 in Border, K., & Jordan, J. (1983). Straightforward elections, unanimity and phantom voters. The Review of Economic Studies, 50(1), 153–170. Retrieved from http://restud.oxfordjournals.org/content/50/1/153.short. I omit the context of the proof to concentrate on the particular argument.
- We assume $\infty<p_l<p^\circ$ (all symbols are real numbers).
- At some point we have proved that:
- for any $0<\epsilon^\circ < \frac{(p^\circ-p_l)}{2}$,
- for any $p^*$ with $2p_l-p^\circ + \epsilon^\circ < p^* < p_l-\epsilon^\circ$
- we have $c(p^*) \in [2p_l-p^\circ-2\epsilon,~2p_l-p^\circ +\epsilon]$ for all $0<\epsilon<\epsilon^\circ$.
From there, the authors conclude that $c(p) = 2p_l - p^\circ$ for all $p\in [2p_l-p^\circ,p_l)$.
Notice that the authors are very clear on the fact that they do not assume that $c(\cdot)$ is continuous:
"If we were to impose continuity as a requirement on C (and hence c) then this could not happen [...]. We have not imposed continuity in the interest of avoiding mathematical assumptions which might cloud the interpretation of our results."
Given that $c(\cdot)$ is not necessarily continuous, I am puzzled by the argument. It is clear that, as $\epsilon^\circ$ tends to zero, $c(p^*)$ gets arbitrarilly close to $2p_l - p^\circ$, but without assuming that $c(\cdot)$ is continuous, how can we be sure that $c(\cdot)$ does not exhibit a jump at $\epsilon=0$?
As I was writing the question, I understood my mistake and I figured out I might share it in case it could help someone.
Actually, $c(p^*)$ does not vary. What varies is the range of values of $p^*$ for which the property applies, and the range of possible values of $c(p^*)$.
So we have that, as $\epsilon^\circ$ tends to 0, $p^*$ is contained in a smaller and smaller interval that tends to $(2p_l-p^\circ,p_l)$ (not clear to me why the lower-bond is included in the interval in the paper...). What you need to go from "$p^* \in (2p_l-p^\circ+\epsilon^\circ,p_l-\epsilon^\circ)$ for all $\epsilon^\circ>0$" to "$p^* \in (2p_l-p^\circ+\epsilon^\circ,p_l-\epsilon^\circ)$ for all $\epsilon^\circ\geq0$" is the continuity of sequences of open intervals on the real line; not the continuity of $c(\cdot)$.