I was reading the wikipedia article on the Dirac Delta function, and what I saw was that the function needed to have a integral over the complete number line to be $1$, and have a value of zero everywhere except at $0$.
Now my idea is instead of defining the functions as: $$\delta(x)=\lim_{b\rightarrow0}\frac{1}{|b|\sqrt{\pi}}e^{-(\frac{x}{b})^2}$$
Why can’t we take some other function whose value starts at $0$ at negative $\infty$, moves towards $0$ and suddenly changes value to $1$ and remains there? Then we can take the derivative of that function and the derivative would be the Delta function.
Now the function I used was the Sigmoid function. My alternate definition for the Dirac Delta function is: $$\delta(x)\overset{?}{=}\lim_{k\rightarrow{+\infty}}\frac{d}{dx}\sigma(kx)$$
It’s graph suggests that the definition might be correct. Am I right in thinking that this can be a suitable alternate definition?