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While learning a little Fourier analysis, I ran into this interesting phenomenon:

Consider a series of sawtooth waves such that the height and width of the sawteeth shrinks to zero, but the slope of the sawteeth remains the same. To be specific, let

$f_n(x) = \frac{nx - \lfloor nx\rfloor}{n}$

Then define

$F(x) = \lim_{n\to\infty}f_n(x)$

It seems intuitively clear that $F(x) = 0$ for all $x$ because the global maximum of $f_n$ is $\frac{1}{n}$.

If $F(x) = 0$, then we should have F'(x) = 0 as well. However, if we choose an irrational value of $x$, then f'_n(x) = 1 for all $n$, so if F'(x) is found instead by taking

F'(x) = \lim_{n\to\infty}f'_n(x)

we do not get F'(x) = 0.

It seems like the derivative of a limit is not the same as the limit of a derivative, which is pretty counterintuitive to me.

What's going on?

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    You may want to have a look at this as well, although a different, you might find it interseting. http://en.wikipedia.org/wiki/Uniform_convergence under the section 'Applications'. It talks about the differentiability and the derivative of the limit function under pointwise convergence.2011-09-16

1 Answers 1

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You say that you expect to be able to interchange limit and derivative operations. Now, derivative itself includes a limit operation, so I wonder whether you expect to be able to interchange any two different limit operations. If so, this discussion by Tim Gowers might be helpful.