If $f(x) = |x|\mathbf{1}_{[-1,2]}(x)$, $x \in \mathbb{R}$, then
- $(T_f)' = \delta_{-1} - 2\delta_2$
- $(T_f)' = \delta_{-1} + \mathbf{1}_{[-1,2]}(x)\operatorname{sgn}(x) - 2\delta_2$
- $T_f$ is not differentiable in $\mathcal{D}'(\mathbb{R})$
- $(T_f)' = \delta_{-1} - \delta_2$
I can't understand the notation. What are $T_f$ and $(T_f)'$?
Solving the exercise it just seems that $T_f$ is the signal itself and $(T_f)'$ its derivative in the sense of distributions. Why not just calling them that way then?
Distributions are continuous linear functionals on the space of test functions $\mathcal{D}$. Then each locally integrable function $f$ is identified as a distribution via the inclusion
$$ L^1_{\text{loc}} \ni f \mapsto T_f \in \mathcal{D}', \qquad T_f(\varphi) = \int f\varphi. $$
Although we often deliberately abuse the notation to write $f$ in place of $T_f$, technically they are different types of objects and one needs to distinguish them whenever necessary.