What is the integral of the Dirac delta function between these limits?

Question

I've learnt that $\displaystyle\int_{-k}^k\delta(x)dx=1$ where $k$ is any positive real number. But what is $\displaystyle\int_{-k}^0\delta(x)dx$?

Answer 1

The integral of dirac delta is the heaviside step function, $H(x)$.

The step function is basically $0$, but if $x$ is postive, then $1$.

Using half-maximum convention, the integral would be $\frac{1}{2}$.

Using regular convention, just $0$, because $H(0)-H(-k)=0-0=0.$

So it depends on what value of $H(0)$ you use.