Delta function integrated from zero

Solution 1:

The Wikipedia formula is only valid for $a>0$, but not for $a<0$ or $a=0$.

The left hand side of their formula makes sense, however, and equals zero when $a<0$ and equals one-half (as you expect) when $a=0$.

It may be easier to understand by rewriting the integral as $$ \int_{-\infty}^\infty \delta(r-a)\ e^{-s r}\ \mathbb{I}_{[0,\infty)}(r) \ dr$$ where $\mathbb{I}$ is the indicator function of the positive half of the line. If you treat the delta function above as a limiting case of even functions peaked at zero, you will get the result for any value of $a$.

Solution 2:

The equation you quote from Wikipedia (from this section) specifies a Laplace transform, and the article on the Laplace transform (in this section) states that the intended meaning of that integral is the limit as $0$ is approached from the left.