Let $r:[0,1]\to\mathbb R$ be a continuous function and let $u_\lambda$ be the unique solution of the Cauchy Problem:
\begin{cases}u''(t)+\lambda r(t)u(t)=0,\quad\forall t\in [0,1],\\ u(0)=0,\quad u'(0)=1.\end{cases}
It is well known that $(u_{\lambda_n})$ converges uniformly on $[0,1]$ to $u_\lambda$ whenever $\lambda_n\to\lambda.$ Define the map $\tau:\mathbb R\to\mathbb R$ by setting
$\tau(\lambda):=\inf\{t\in(0,1]\mid u_\lambda(t)=0\},$ with the convention $\tau(\lambda)=1$ if $u_\lambda(t)\neq 0$ for every $t\in (0,1]$. Prove then that $\tau$ is continuous.
Edit:
ok so i've been trying to work on Robert hint. I am trying to prove that if $\tau(\lambda_0)=\tau_0<1$ then for some $\varepsilon>0$ small i must have $u_{\lambda_0}(\tau_0+\varepsilon)<0$ , but i seem to go nowhere farther from some silly tryings using mean value theorem or stuff like that.. Where am i missing the key point?