# Tagged Questions

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

494 views

43 views

104 views

### An analytic family of in fact non-existent improper Riemann integrals

Question: Are there any useful interpretations or "applications" of the formula $$\intop_0^\infty e^{-ax}\frac{\sin x}{x}\,dx=\frac{\pi}{2}-\arctan(a)\qquad \forall\;a\in \mathbb{R},$$ in which the ...
139 views

142 views

### separable BV space for PDE's, Whats stopping us? [closed]

Consider the metric space BV(0,1) with the following metric $$d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)|$$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing ...
238 views

### Fourier transform localisation (still unanswered, but apparently off-topic?) [closed]

In the context of Pólya's theorem I was reading these notes here on p. 19. In the last paragraph the authors claim (it is the sentence starting like "standard Fourier theory shows...") that the ...
Suppose a real differentiable function $h(x)$ with its derivative not infinity, it is sure that its second symmetric derivative $\lim_{\epsilon->0}\frac{h(x+\epsilon)-2h(x)+h(x-\epsilon)}{\epsilon^... 0answers 72 views ### Derivatives of Mollified functions I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following: Let$\sigma(t,x)$be a matrix of dimension$d\times d$, and let$b(t,x)$... 0answers 152 views ### Improving Baumgartner's result? Q1: Is it consistent with the failure of CH to have an$\aleph_1$-dense subset$A \subseteq \mathbb{R}$such that for every$X \subseteq \mathbb{R}$of size$\aleph_1$, there is a$C^{\infty}$map$F: ...
Question. Let $V_1,\ldots,V_n$ be open, bounded and convex subsets of $\mathbb R^2$. Show that $F=\mathbb R^2\smallsetminus\bigcup_{i=1}^n V_i$ possesses only finitely many connected components. I ...