Proposition: Conditional critical-root stability

Suppose two response functions are continuously differentiable on one interval, each has a unique interior root at level $q_{\ast }$, and each satisfies ${\partial }_{\beta }Q\le -c_0<0$. If their sup-norm difference is at most $ϵ$, the roots differ by at most $ϵ/c_0$.

Proof. Evaluate one response at the other's root. The deviation from $q_{\ast }$ is at most $ϵ$. Apply the mean-value theorem and the derivative lower bound in absolute value.