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\text { Let } f: \mathbf{R} \rightarrow \mathbf{R} \text { be a Lipschitz function, that is, } \sup \left\{\frac{|f(x)-f(y)|}{|x-y|}: x, y \in \mathbf{R} \text { with } x \neq y\right\}<\infty

\text { Suppose that for every } x \in \mathbf{R} \text {, } \lim _{n \rightarrow \infty} n\left[f\left(x+\frac{1}{n}\right)-f(x)\right]=\lim _{n \rightarrow \infty} n\left[f\left(x-\frac{1}{n}\right)-f(x)\right]=0 . \text { Prove that } f \text { is differentiable on } \mathbf{R} \text {. }

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