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?