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For an explicit example, you might try $$ f(x) = \sum_{n=1}^\infty \dfrac{1}{n^2} (\tanh(n^2(x-n)) - 1)$$$$ f(x) = \sum_{n=1}^\infty \dfrac{1}{n^2} (1 - \tanh(n^2(x-n)))$$ noting that $f'(n) < -1$.
For an explicit example, you might try $$ f(x) = \sum_{n=1}^\infty \dfrac{1}{n^2} (\tanh(n^2(x-n)) - 1)$$ noting that $f'(n) < -1$.
For an explicit example, you might try $$ f(x) = \sum_{n=1}^\infty \dfrac{1}{n^2} (1 - \tanh(n^2(x-n)))$$ noting that $f'(n) < -1$.
