At least according to what I see in mathematica, $\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\delta(t^{\prime\prime}-t_{1})+\int_{t^{\prime}}^{t}dt^{\prime\prime}\delta(t^{\prime\prime}-t_{1})\neq\int_{t_{0}}^{t}dt^{\prime\prime}\delta(t^{\prime\prime}-t_{1})$
By assuming $t_{0}=0, t>0$ for simplicity, $\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\delta(t^{\prime\prime}-t_{1})+\int_{t^{\prime}}^{t}dt^{\prime\prime}\delta(t^{\prime\prime}-t_{1})=2\theta(t_{1})-2\theta(t_{1}-t)+\theta(t_{1}-t)\theta(t-t_{1})$ while $\int_{t_{0}}^{t}dt^{\prime\prime}\delta(t^{\prime\prime}-t_{1})=\theta(t_{1})\theta(t-t_{1})$, where $\theta$ is the Heaviside theta function.
However, one might want to claim that $\delta(t^{\prime\prime}-t_{1})=\frac{1}{\pi}\lim_{\epsilon\rightarrow0}\frac{\epsilon}{(t^{\prime\prime}-t_{1})^{2}+\epsilon^{2}}$ and then, $\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\lim_{\epsilon\rightarrow0}\frac{\epsilon}{(t^{\prime\prime}-t_{1})^{2}+\epsilon^{2}}+\int_{t^{\prime}}^{t}dt^{\prime\prime}\lim_{\epsilon\rightarrow0}\frac{\epsilon}{(t^{\prime\prime}-t_{1})^{2}+\epsilon^{2}} =_{?}\lim_{\epsilon\rightarrow0}\left(\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}\frac{\epsilon}{(t^{\prime\prime}-t_{1})^{2}+\epsilon^{2}}+\int_{t^{\prime}}^{t}dt^{\prime\prime}\frac{\epsilon}{(t^{\prime\prime}-t_{1})^{2}+\epsilon^{2}}\right)=\int_{t_{0}}^{t}dt^{\prime\prime}\delta(t^{\prime\prime}-t_{1})$.
Is the step where the $=_{?}$ sign is a valid one? in other words, do the conditions for interchanging these operations apply in this case?