How to propagate error through a Heaviside step function $H(x\pm\epsilon)$

Question

Suppose we have a set of data points $\{x_1,\cdots,x_n\}$ with a corresponding set of errors $\{\epsilon_1,\cdots,\epsilon_n\}$. Using the standard definition of the Heaviside step function, $$H(x)=\cases{0\ , x<0\\1\ , x\geq 0},$$ how would one propagate the error in $x$ when calculating $C$? $$C=\sum_{i=1}^n{H(x_i)}$$ In other words, how does one handle errors when dealing with Heaviside functions $H(x\pm \epsilon)$? I am at a loss for the correct approach, as the typical means of error propagation would involve a Dirac delta function $\delta(x)$. To see this consider, $\delta C=|H'(x)|\delta x$, where $$H'(x)=\delta(x)=\cases{\infty\ , x=0\\0\ ,x\neq0 }$$

Answer 1

There is no sense of "propagation" when you are using discontinuous functions. Propagaiton makes sense when there is a smooth relationship between input errors and output errors.

On one hand, your $C$ is quite robust to errors if $|x_i|\gg 0$. If not, then you need to consider a mixed bernoulli model, which will require a monte carlo simulation. Basically, you want to model each Heaviside function as a coin flip, where $P(H_i)=P(x_i + \epsilon_i > 0)$ and look at the distribution of the sum, $C$.