On the Roots of a Continuous Non-Differentiable Function

Let f (x) be a continuous non-differentiable function defined in the interval (0,1), whose upper and lower bounds are l and u, say. If l ≤ α ≤ u, we denote by S(α), the set of points x in (0, 1) for which f(x)=α. It is known that S(α) is closed and non-dense. We now divide the interval (l, u) into three distinct sets of points A, B and C, so that A+B+C = (l, u), and

(a) A = the set of points a for which is of positive measure

(b) B = the set of points α for which S(α) is non-enumerable but of measure zero

(c) C = the set of points α for which S(α) is enumerable.