Find integral limits for step functions I have an equation of the form
$$f(t) + \int_0^t H(t') dt' = c$$.
For now, assume that $f(t)$ is linear, and $H(t)$ is some arbitrary step function (not necessarily just a standard heaviside step function. Is there any easy/ logical way of solving this to find the integral limit t. It feels like it should be easy, as the integral of the step function is 'just' a piecewise linear function, but I'm really struggling to do this.
While I don't have any specific values in mind, I am most interested in the following:

*

*The values of the n components of H(t) form a non-negative decreasing sequence.


*$f(t) = kt$, where $k > 0$.


*$c > 0$.
Under these conditions, I think there should be a unique solution (though would be interested to hear if I am wrong in assuming this!).

For any function $h$ (not necessarily step),
$$f(t)+\int_0^t h(t)\,dt=c$$
is
$$f(t)+H(t)-H(0)=c$$ where $H$ is an antiderivative of $h$. To obtain $t$, you solve this ordinary, univariate equation.
If $h$ is a step function, its antiderivative is piecewise linear and continuous. You can solve the equation in every piece independently (and check that the solution(s) belong to the definition interval of the piece).
If $f$ is $kt$, then $t=0$ is always a solution. There can be a second solution if the values of $h$ are larger, then smaller than $k$.
