We are given $F(x) = |2x+1|, x \in \mathbb{R}$
How to determine whether $$[F|_{(0, \infty)}] \in \mathcal{D}'((0, \infty))$$
$$[F|_{(- \infty, 0)}] \in \mathcal{D}'((- \infty, 0))$$
are regular distributions?
When it comes to the first one, we have $F|_{(0, \infty)}(x) = 2x+1$ and this function is continuous, so it is locally integrable, so it induces a regular distribution.
But when it comes to the second one, we have that on interval $(- \infty, - \frac{1}{2})$ , $F(x) = -2x-1$ and on $[ - \frac{1}{2}, 0)$, $F(x) = 2x+1$.
So we need to determine whether in the second case, we can write down the distribution as
$$<F, \varphi> = \int_{\mathbb{R}} F(x) \varphi (x) dx $$ for a test function $\varphi \in D(\mathbb{R})$, that is a smooth function with compact support.
Could you help me with that?