Given a function $f\in L^{\infty}([0,1],\mathbb{R}^d)$. If a consider the following Cauchy problem:
$$ \dot{y}(t) = f(t), \quad \text{for almost every } t \in [0,1]. $$ I want to understand, why often we do not speak of the value of $f$ at each time instant $t \in [0,1]$.
For example, in this case, we can't write $f(\frac{1}{2})$. And from what I have seen in some textbooks, that it requires to use the notion of Lebesgue point in order to write $f(a)$ (with $a$ is a Lebesgue point).
Can we also say that $f$ is defined (only) almost everywhere?
Definition of a Lebesgue point: https://fr.wikipedia.org/wiki/Point_de_Lebesgue