text: Q:
  
  About the continuity of the Fourier transform.
  
  If $ x^{\alpha} g \in L^1 ( \Bbb R^n)$ for $| \alpha | \leqslant k$, then how can I prove that its Fourier transform $$  \mathscr{F} g \in C^k ( \Bbb R^n) ?$$ Here $\alpha$ is a multi-index.
  
  A:
  
  You know that if $g\in L^{1}(\mathbb{R}^{n})$ then $\mathscr{F}g\in C(\mathbb{R}^{n})$
  Now try to prove this formula: $$x^{\alpha}D^{\beta}_{x}\mathscr{F}f(x)=(-1)^{|\beta|}\mathscr{F}(D^{\alpha}x^{\beta}f)(x)$$
  Then conclude
  
  