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IMC 1998 Problem 12

Source: IMC 1998 Day 2 Problem 6

October 28, 2020
differentiabilityreal analysis

Problem Statement

f:(0,1)→[0,∞)f: (0,1) \rightarrow [0, \infty) is zero except at a countable set of points a1,a2,a3,...a_{1}, a_2, a_3, ... . Let bn=f(an)b_n = f(a_n). Show that if ∑bn\sum b_{n} converges, then ff is differentiable at at least one point. Show that for any sequence bnb_{n} of non-negative reals with ∑bn=∞\sum b_{n} =\infty , we can find a sequence ana_{n} such that the function ff defined as above is nowhere differentiable.
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