The dirac delta can be defined in such a way that
$$ \int_{-\infty}^{\infty}\delta(z)\,dz = 1 $$
and, for a suitable $f$, it holds that
$$ \int_{-\infty}^{\infty}f(z)\,\delta(z)\,dz = f(0). $$
What can be said about the integral (take $a>0$)
$$ \int_0^{a}f(z)\,\delta(z)\,dz\quad? $$
Is still true that $\int_0^{a}f(z)\,\delta(z)\,dz=f(0)$ ? I am pretty sure that, for all $\varepsilon>0$, then
$$ \int_{-\varepsilon}^{\varepsilon}f(z)\,\delta(z)\,dz=f(0), $$
but I am in trouble when the interval of integration has the zero at its boundary.
With 0 on the boundary it is a priori not defined. It corresponds to integrating $\delta$ against the function $f 1_{x>0}$ (or possibly $f 1_{x\geq 0}$) which is not continuous at $0$ where the support of $\delta$ lies. Note that the other case $(-\epsilon,\epsilon)$ presents no problem since $\delta$ has support in 0.