Let $\Omega$ be a convex, open subset of $\mathbb{R}^d$, with a smooth boundary. Consider the set $$M = \{f/f:\Omega\to\mathbb{R} \bigwedge f\in C^0(\Omega)\bigwedge \lvert| f|\rvert_{W^{1,d+1}}\in\mathbb{R}\}$$ From Morrey's inequality, $f\in M \implies f\in C^{0,1/(d+1)}(\Omega)$. Lets take an...

Here's a fun proof that I made in this answer: Consider the more basic power rule: $$\int_1^xt^{n-1}\ dt=\frac{t^n}n\bigg|_1^x=\frac{x^n-1}n$$ Of course, we can't simply have $n=0$, but if we take the limit as $n\to0$, you end up with $$\int_1^xt^{-1}\ dt=\lim_{n\to0}\frac{x^n-1}n$$ And sinc...

There are several approaches. One is based on the limit $$ \lim_{n\to\infty}n\left(x^{1/n}-1\right)=\log(x)\tag{1} $$ By adjusting the constant of integration, we have $$ \int x^n\,\mathrm{d}x=\frac1{n+1}\left(x^{n+1}-1\right)+C\tag{2} $$ Taking the limit of $(2)$ as $n\to-1$ and using $(1)$ yiel...