We can pick any non-zero $x$ and get zero. Let's says we use Paul Garrett's solution: $H(x) \cdot H(-x)$. Thank you very much to Thomas Andrews and Paul Garrett for the solutions they provided!īriefly: one can use either multiplication or addition/subtraction to manipulate the Heaviside function in order to get it nonzero only at one point. That works, of course, for test functions only.ĭisclaimer: honestly speaking, it feels to me like I'm doing some inappropriate things to mathematics here. Therefore, I conclude that the distributional derivative of my $S(t)$ function is the Dirac delta function as well and I'm good to go. But my function $S(t)$ is also constant on the interval $(-\epsilon, \epsilon )$ and I can also take it out from the integral.
That procedure makes use of the fact that the Heaviside function $H(t)$ is constant on the interval $[0, \infty )$ and one can take it out from the integral. To do that, I make use of the procedure of proving that the Dirac delta function is the distributional derivative of the Heaviside function. I have to find out the distributional derivative of my $S(t)$ function. I know that distributional derivative of the Heaviside function is the Dirac delta function. Then I want to take a derivative of that expression. Then I write down my piecewise function as: Therefore, I decided that I had to introduce a function similar to the Heaviside function but that looks like this: I decided that Kronecker delta wasn't a good fit because, first, it can take only two arguments, second, we normally use it with natural numbers only. The closest I could find was Kronecker delta. Therefore, I checked if there are functions out there that can represent a single point.
I don't see a way to use the Heaviside function to represent a single point. Then how do I write it down using the Heaviside function? QUESTION: if I have a piecewise function that look like this: Thus, if I a have a function that looks like this:
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Here and here I saw how to rewrite a piecewise function using the Heaviside function.