The Mikhlin multiplier theorem states the following:
Theorem [Theorem 2, Davide Guidetti Vector valued Fourier multipliers and applications]. Let $m$ be a bounded function on $\displaystyle \mathbb{R}^{n}$ which is smooth except possibly at the origin, and such that the function ${\displaystyle |x|^{k}|\nabla ^{k}m|}$ is bounded for all integers ${\displaystyle 0\leq k\leq n/2+1}$: then $m$ is an $L^p$ multiplier for all $1 < p < \infty$.
Why not considered at the origin? At origin is trivial? (Because if $x=0$ then ${\displaystyle |x|^{k}|\nabla ^{k}m|}$ is bounded automatically?