Weak derivative of this function

Question

What is the weak derivative of this function:

$(2x-a)^{1/2}1_{2x\geq a}$.

Answer 1

Let $f$ be a smooth function on $\mathbb{R}$ of compact support. Compute \begin{align*} \int_{\mathbb{R}}(2x-a)^{1/2}1_{2x\ge a} \ f'(x)\ dx & = \lim_{\varepsilon\to 0+}\int_{a/2+\varepsilon}^\infty (2x-a)^{1/2}f'(x)\ dx \\ & = \lim_{\varepsilon\to 0+}(-(2\varepsilon)^{1/2} f'(a/2+\varepsilon)-\int_{a/2+\varepsilon}^\infty (2x-a)^{-1/2}f(x)\ dx) \\ & = -\int_{a/2}^\infty (2x-a)^{-1/2}f(x)\ dx. \end{align*} This implies that the weak derivative of $(2x-a)^{1/2}1_{2x\geq a}$ is $(2x-a)^{-1/2}1_{2x\geq a}$.