Verify and show that the following is a distribution and prove if it is a tempered distribution

Question

I've been struggling to verify and show that the following function is a distribution and whether its a tempered distribution.

Let $f(x)=\frac{\cos x}{\sqrt{|x|}}$ for $x \in \mathbf{R}$.

Really do appreciate the help and any insight on how to solve the problem. Thanks!

Answer 1

1. A locally integrable function induces a distribution. $|x|^{-1/2}$ is locally integrable.
2. Multiplication of a distribution with a smooth ($C^\infty$) function gives a distribution. $\cos x$ is a smooth function.
3. A distribution that doesn't grow too quickly at infinity is a tempered distribution. $|x|^{-1/2} \cos x$ doesn't grow at all at infinity.