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It is shown, by using properties of a symmetric correlated random walk, that almost all level sets of $f_r$ are finite (with respect to Lebesgue measure on the range of $f$), but that for an abscissa $x$ chosen at random from $[0,1)$, the level set at level $y=f_r(x)$ is uncountable almost surely. 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Allaart","submitted_at":"2013-12-07T17:54:39Z","abstract_excerpt":"This paper examines the level sets of the continuous but nowhere differentiable functions \\begin{equation*} f_r(x)=\\sum_{n=0}^\\infty r^{-n}\\phi(r^n x), \\end{equation*} where $\\phi(x)$ is the distance from $x$ to the nearest integer, and $r$ is an integer with $r\\geq 2$. It is shown, by using properties of a symmetric correlated random walk, that almost all level sets of $f_r$ are finite (with respect to Lebesgue measure on the range of $f$), but that for an abscissa $x$ chosen at random from $[0,1)$, the level set at level $y=f_r(x)$ is uncountable almost surely. 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