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• 10:40, 12 November 2022 Palimpseste talk contribs created page Improper Integral of Partial Derivative (Created page with "==Theorem== <onlyinclude> Let $I$ and $U$ be open intervals of $\R$, and $f: U \times I \to \R$ be a function such that: # for all $u \in U$, $f(\cdot,u) : I \to \R$ is integrable; # for all $t \in I$, $f(t,\cdot) : U \to \R$ is $\CC^1$; # there is an integrable function $\phi : I \to \R$ such that for all $(t,u) \ in I \times U$, :$\ds \size {\partial_u f(t,u)} \leq \phi(t)$ Then the function $F : U\to \R$ defined by :$\ds F(u) = \int_I f(t,u) \rd t$ is $\CC^1$ and :$\...") Tag: Visual edit: Switched
• 20:36, 6 November 2022 Palimpseste talk contribs created page Talk:Dirichlet Integral/Proof 2 (Created page with "How do you see that $I$ is actually continuous at 0 from the comparison theorem ? It seems to me that $\dfrac{\sin x}{x}$ is far from being integrable on $(0;+\infty)$... In fact, I believe there are two things to prove in addition to what is there: 1. prove that $\ds\int_0^{+\infty}\frac{\sin x}{x} \rd x$ actually has a finite value; 2. prove that $\ds I \to_{0}\int_0^{+\infty}\frac{\sin x}{x} \rd x$. The first point is more or less straightforward (basically the sa...")