Let $B$ denote Brownian motion started at 0. Given intervals $I_1=[a_1,b_1],..., I_n = [a_n,b_n]$ with $0<a_1,b_1<a_2$, $b_2<a_3$, ... $b_{n-1}<a_n$, let $y\geq 0$ and define
$$f(y) = P(\textrm{There exist } t_i\in I_i\textrm{ so that } B(t_i) = y\textrm{ for all } i).$$
My question is, is $f$ a decreasing function? To me, it seems like it should be true... But I have not been able to prove or disprove it. Any help would be appreciated.