How does the Dirac delta function operate when its peak is at the boundary of an integral?

Question

As far as I can tell the Dirac delta function in an integral picks the value of the multiplying function at the peak provided the peak is within the boundary, i.e. $$\int^{a+e}_{a-e} \delta (x-a) f(x)\ \mathrm{d}x = f(a).$$ But what happens if a peak is a boundary? $$\int^{a}_{a-e} \delta (x-a) f(x)\ \mathrm{d}x$$ My attempt is $$\lim_{u\to 0}\int^{a+u}_{a-e} \delta (x-a) f(x)\ \mathrm{d}x = \lim_{u\to 0}f(a)$$ but while this process is reasonably rigorous for a function with a primitive, it's not necessarily true for this.