\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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