# 2015-21 Differentiable function

Assume that a function $$f : (0, 1) \to [0, \infty)$$ satisfies $$f(x) = 0$$ at all but countably many points $$x_1, x_2, \cdots$$. Let $$y_n = f(x_n)$$. Prove that, if $$\sum_{n=1}^{\infty} y_n < \infty$$, then $$f$$ is differentiable at some point.

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