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On the Roots of a Continuous Non-Differentiable Function
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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.
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