Let $\delta$ be the Dirac distribution on $\mathbb{R}$. Find functions $f_{0}, f_{1} \in L^{2}(\mathbb{R})$ such that $$ \delta=f_{0}+\frac{d f_1}{d x} $$
Note: The derivative given above is in the distributional sense. For every test function $\phi \in \mathcal{D}(\mathbb{R}) = C_c^{\infty}(\mathbb{R})$. The above relation will satisfy $$\delta(\phi) = \phi(0) = \int_{\mathbb{R}}f_0(x)\phi(x)dx+\int_{\mathbb{R}}f_1(x)\phi^{\prime}(x)dx, \,\, \forall \,\, \phi \in C_c^{\infty}(\mathbb{R}) $$ I am not sure how to proceed from here, many techniques that are used to manipulate integrals require the integrand to be at least continuous. Any hints?