I know that the characteristic function $1_{\mathbb{Q}}$ is a Borel function (from $\mathbb{R}$ to $\mathbb{R}$), but I wonder if this function actually appears "naturally" in applications (and is not just an exotic example to show that borel functions are more general than piecewise continuous function).
So my question is this : are there explicit Borel functions (and even $L^1(\mathbb{R}))$ that are non-piecewise continuous, but still appear as fundamental or very important in applications (in analysis, or any other domain of mathematics or physics) and are not simply exotic objects?
(I am aware it's not a very well-defined question, but I hope it's understandable)
My answer will be a bit indirect.
Most mathematicians and physicists never use exotic functions such as $1_{\mathbb{Q}}$, except as counter-examples. However, we want our function spaces such as $L^1$ and $L^2$ to be large enough. Basically we want that "reasonable" sequences of functions have limits, which translates into the concept of Banach space. The fact that $L^1$ is a Banach space allows us to use a whole machinery of strong theorems. In applications, quantum mechanics depends heavily on the fact that the "space of states" is a Hilbert space (complete with inner product).
If we were to limit ourselves to integrate piecewise continuous, or even Riemann-integrable functions, then the resulting function space wouldn't be a Banach space. So we definitely want these functions to exist, they're useful as a whole even though each individual exotic function is hardly ever used.