Let $f$ be an increasing and continuous real function with exactly one root. Let $\tau=\min\left\{k \in \mathbb{Z}, f(k) \geq 0\right\}$ be the first integer such that f is nonnegative (i.e. the first integer above the root). Now, let $f_n$ be a sequence of increasing, continuous real functions that converge pointwise towards $f$. I would like to find a sequence with discrete values $\tau_n$ defined only from $f_n$ such that $\tau_n \to \tau$.
It is natural to consider $\tau_n=\min\left\{k \in \mathbb{Z}, f_n(k) \geq 0\right\}$. However the convergence of $\tau_n$ towards $\tau$ is not guaranteed : for $f(x)=x, f_n(x)=x-1/n$, we have $\tau=0$, and $\tau_n=1$ for every $n$. Is there a way to circumvent this problem?