/sci/ - Science & Math

>>12473113
We define [math]\chi _n : \mathbb{R} \rightarrow \mathbb{R}[/math] by [math]\max(n - n^4x, 0)[/math] if [math]x \geq 0[/math] and [math]\max(n + n^4 x, 0)[/math] if [math]x < 0[/math]. Unless I've done something wrong:
>[math]\chi _n[/math] is continuous
>[math]\int _{- \infty}^{\infty} \chi _n (x) \ dx = \frac{1}{n^2}[/math].
>[math]\chi _n[/math] has support in [math][-0.5, 0.5][/math] for [math]n \geq 2[/math]
Then, by setting [math]f(x) = \sum _{i = 2}^{\infty} \chi _n (x - n)[/math], we compute that [math]\int _{- \infty}^{\infty} f(x) \ dx = \sum _{n=2}^{\infty} \frac{1}{n^2}[/math], which is finite. Plus, since the components all have disjoint supports, we get to keep continuity.
Then you fuck around with some computations and conclude that [math]G(x)[/math] can't be Lipschitz because [math]f[/math] gets too fucking high (left to reader).
IIRC for continuous functions that's equivalent to boundedness of [math]f[/math].

>>11595648
>hey lad whachu doing
>ah, I'm just trying to put out this fire by throwing gasoline at it