TITLE: Is there a continuous function from $[0,1]$ to $\mathbb R$ that satisfies
QUESTION [18 upvotes]: Is there a continuous function $f:[0,1] \to \mathbb R$ such that $f(x) = 0$ uncountably often and, for every $x$ such that $f(x) = 0$, in any neighbourhood of $x$ there are $a$ and $b$ such that $f(a) > 0$ and $f(b) < 0$?

REPLY [6 votes]: In addition to the carefully constructed examples others have given, I just want to note that with probability $1$ one-dimensional Brownian motion starting at $0$ has these properties.