This question might be related a little bit to physics, but wanted to hear the answer from mathematics perspective and mathematician's intuition.
Imagine we have an object that moves in the time interval of $[t_0, t_1]$ and has the following trajectory : $x(t) + \epsilon\theta(t)$ where $\epsilon$ is infinetisemal number and $\theta(t)$ is given such as:
$ \theta(t)= \begin{cases} 1&t_0 < t < t_1\\ 0&\text{elsewhere} \end{cases} $
Now, let's say I want to figure out the velocity. So we derivate and get: $\dot x(t) + \epsilon \dot \theta(t) = \dot x(t) + \epsilon \delta(t-t_0)$.
I am trying to understand one thing now. So at $t_0$, object was at $x(t_0)$ and what was its speed at $t_0$ ? I get: $\dot x(t_0) + \epsilon \infty$, because at $t_0$, my delta function is infinity, so what is the speed at $t_0$ really ? Is it infinite ? but since $\epsilon$ is infinetisemal, multiplying it with infinity doesn't give me the intuitive answer what the speed is at that moment.
Firstly, note that you forgot the second boundary of the "bump", hence formally $\dot{\theta}(t) = \delta(t-t_0) - \delta(t-t_1)$.
Now, the question is a bit awkward, because we can consider such a quantity $x(t)$ mathematically of course, but it cannot really represent a position anymore, at least because of its distributional nature, which is unphysical.
You may fix this situation by arguing that the infinity behind the Dirac delta is compensated by the infinitesimality of $\epsilon$, in other words, by setting something like $\epsilon \cdot \infty = 1$, but it is quite unnatural.
Physics, as well as science in general, is about choosing the right mathematical model in order to describe physical phenomena, which are seldom discontinuous (a professor of mine used to say jokingly that Nature was twice continuously differentiable). Sure, we can construct models with approximations, but they should be justified and not bring unnecessary mathematical illnesses on top of that.
In the present case, a more reasonable model would replace the discontinuities of $\theta$ at $t_0$ and $t_1$ by some sharp sigmoids and the delta function would be replaced by some narrow gaussian for instance.