The distributional derivative of a piecewise $C^1$ function with a jump: $(u_p{)}^{\mathrm{\prime }}=u_{p_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}}+[p]{\delta }_{t_{\ast }}$

Local integrability: put $I_{-}:=I\cap (-\mathrm{\infty },t_{\ast })$ and $I_{+}:=I\cap (t_{\ast },\mathrm{\infty })$. Both are open, and $p$ and $p_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}$ are continuous on each. For any open $O\subseteq \mathbb{R}$, the preimage on either half is open relative to that open half, hence open in $\mathbb{R}$; the preimage at $t_{\ast }$ is either empty or $\{t_{\ast }\}$. Thus both functions are Borel on $I$. Let $K\subseteq I$ be compact. If $K=\mathrm{\varnothing }$, its integral is zero. Otherwise choose $c<t_{\ast }<d$ with $K\cup \{t_{\ast }\}\subset (c,d)$ and $[c,d]\subset I$. Extending each one-sided reading of $p$ and $p_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}$ to $t_{\ast }$ by its finite one-sided limit gives continuous functions on $[c,t_{\ast }]$ and $[t_{\ast },d]$; [L7] bounds them there, and the fixed finite values at $t_{\ast }$ preserve boundedness. Hence $p$ and $p_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}$ are bounded on $[c,d]$, and their absolute values are Borel and continuous except possibly at $t_{\ast }$. By [L3] they are Riemann integrable and Lebesgue integrable on $[c,d]$. The set $K$ is Borel by [L5], and [L6] gives ${\int }_K\mathrm{\mid }p\mathrm{\mid }d\lambda \le {\int }_{[c,d]}\mathrm{\mid }p\mathrm{\mid }d\lambda <\mathrm{\infty }$ and ${\int }_K\mathrm{\mid }{p}_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}\mathrm{\mid }d\lambda \le {\int }_{[c,d]}\mathrm{\mid }{p}_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}\mathrm{\mid }d\lambda <\mathrm{\infty }$. Thus both functions are locally integrable and define the regular distributions in [L1].

Let $\phi \in \mathcal{D}(I)$ be real-valued. Choose $a<t_{\ast }<b$ with $\mathrm{supp}\phi \cup \{t_{\ast }\}\subset (a,b)$ and $[a,b]\subset I$; compactness of the support and openness of $I$ allow this, without requiring a symmetric interval around $t_{\ast }$. Since $\phi$ vanishes outside $[a,b]$, [L1] and [L3] give ${\int }_{I}p{\phi }^{\mathrm{\prime }}d\lambda ={\int }_{[a,b]}p{\phi }^{\mathrm{\prime }}d\lambda ={\int }_{[a,t_{\ast }]}p{\phi }^{\mathrm{\prime }}d\lambda +{\int }_{[t_{\ast },b]}p{\phi }^{\mathrm{\prime }}d\lambda ,$ the second equality splitting the integral over the two overlapping half-intervals by linearity and the identity ${\chi }_{[a,b]}={\chi }_{[a,t_{\ast }]}+{\chi }_{[t_{\ast },b]}-{\chi }_{\{t_{\ast }\}}$ (the subtracted term contributes $0$ because the singleton $\{t_{\ast }\}$ has measure zero).

On the left half-interval define ${\bar{p}}_{-}(t):=p(t)$ for $t<t_{\ast }$ and ${\bar{p}}_{-}(t_{\ast }):=p(t_{\ast }^{-})$; on $[a,t_{\ast }]$ this extension is continuous. For $t<t_{\ast }$, the mean value theorem [L4] on $[t,t_{\ast }]$ gives $({\bar{p}}_{-}(t_{\ast })-{\bar{p}}_{-}(t))\mathrm{/}(t_{\ast }-t)=p^{\mathrm{\prime }}({\xi }_t)$ for some $t<{\xi }_t<t_{\ast }$. As $t\uparrow t_{\ast }$ this tends to $p^{\mathrm{\prime }}(t_{\ast }^{-})$, proving the endpoint derivative. Its derivative is continuous up to $t_{\ast }$; extend beyond $t_{\ast }$ by the affine function $p(t_{\ast }^{-})+p^{\mathrm{\prime }}(t_{\ast }^{-})(t-t_{\ast })$ to obtain a $C^1$ function on an open neighborhood of $[a,t_{\ast }]$. On $[a,t_{\ast }]$, $p$ and ${\bar{p}}_{-}$ agree except at $t_{\ast }$, so [L3] and [L5] give ${\int }_{[a,t_{\ast }]}p{\phi }^{\mathrm{\prime }}d\lambda ={\int }_a^{t_{\ast }}{\bar{p}}_{-}{\phi }^{\mathrm{\prime }}d\lambda$. Lebesgue integration by parts [L2] for the $C^1$ pair $({\bar{p}}_{-},\phi )$ gives ${\int }_a^{t_{\ast }}{\bar{p}}_{-}{\phi }^{\mathrm{\prime }}={\bar{p}}_{-}(t_{\ast })\phi (t_{\ast })-{\bar{p}}_{-}(a)\phi (a)-{\int }_a^{t_{\ast }}{\bar{p}}_{-}^{\mathrm{\prime }}\phi =p(t_{\ast }^{-})\phi (t_{\ast })-{\int }_{[a,t_{\ast }]}{p}_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}\phi d\lambda ,$ because $\phi (a)=0$ and ${\bar{p}}_{-}^{\mathrm{\prime }}=p_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}$ on $(a,t_{\ast })$; the last integrals agree by [L3] and [L5].

On the right half-interval define ${\bar{p}}_{+}(t):=p(t)$ for $t>t_{\ast }$ and ${\bar{p}}_{+}(t_{\ast }):=p(t_{\ast }^{+})$. The mean value theorem [L4] on $[t_{\ast },t]$ shows that the right derivative of ${\bar{p}}_{+}$ at $t_{\ast }$ is $p^{\mathrm{\prime }}(t_{\ast }^{+})$; extending affinely to the left gives a $C^1$ function on a neighborhood of $[t_{\ast },b]$. Since $p$ and ${\bar{p}}_{+}$, and $p_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}$ and ${\bar{p}}_{+}^{\mathrm{\prime }}$, agree except at $t_{\ast }$, [L3] and [L5] give equality of their integrals against the tests on $[t_{\ast },b]$. Integration by parts [L2], together with $\phi (b)=0$, now gives ${\int }_{[t_{\ast },b]}p{\phi }^{\mathrm{\prime }}d\lambda ={\int }_{t_{\ast }}^b{\bar{p}}_{+}{\phi }^{\mathrm{\prime }}d\lambda ={\bar{p}}_{+}(b)\phi (b)-{\bar{p}}_{+}(t_{\ast })\phi (t_{\ast })-{\int }_{t_{\ast }}^b{\bar{p}}_{+}^{\mathrm{\prime }}\phi =-p(t_{\ast }^{+})\phi (t_{\ast })-{\int }_{[t_{\ast },b]}{p}_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}\phi d\lambda \mathrm{.}$

Adding the two identities of steps 2.1 and 2.2 and substituting into step 1.2 gives ${\int }_{I}p{\phi }^{\mathrm{\prime }}d\lambda =-(p(t_{\ast }^{+})-p(t_{\ast }^{-}))\phi (t_{\ast })-{\int }_I{p}_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}\phi d\lambda =-[p]\phi (t_{\ast })-{\int }_I{p}_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}\phi d\lambda ,$ that is, $⟨(u_p{)}^{\mathrm{\prime }},\phi ⟩=-{\int }_{I}p{\phi }^{\mathrm{\prime }}d\lambda ={\int }_I{p}_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}\phi d\lambda +[p]\phi (t_{\ast })=⟨u_{p_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}}+[p]{\delta }_{t_{\ast }},\phi ⟩$ for every real-valued $\phi \in \mathcal{D}(I)$ by [L1]. Since both sides are complex-linear in $\phi$, the identity extends to all $\phi \in \mathcal{D}(I)$ by applying it to $\mathrm{Re}\phi$ and $\mathrm{Im}\phi$; hence $(u_p{)}^{\mathrm{\prime }}=u_{p_{\mathrm{r}\mathrm{e}\mathrm{g}}^{\mathrm{\prime }}}+[p]{\delta }_{t_{\ast }}$ in ${\mathcal{D}}^{\mathrm{\prime }}(I)$.