What is the Derivative?

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An semi-amusing thing to consider: what is the derivative of this function?

f(x) = \lim_{n \to \infty} \frac{\sin nx}{n}

The standard differentiation process yields cos nx, but at the same time, the value of this function is 0 everywhere − which means it has a derivative of 0. If I remember correctly, the latter is defined to be true. Could be wrong.

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