Suppose I have a "time-sampling" operator given by \begin{align*} S_m: C([0,1]) &\to \mathbb{R}^m \\ f &\mapsto (f(t_1),f(t_2),...,f(t_m)) \end{align*}
Now I want to extend this to $L^2([0,1])$. However, what would happen to the operator if the function was discontinuous exactly at the points $t_i$? For example, consider some piecewise-constant function $f$, whose discontinuities lie exactly at the points $t_i$. Is there a natural way to define what should $f(t_i)$ be?
If such an extension is impossible, then my "gut feeling" says that, at least for piecewise-continuous functions (which is the main case I'm considering), if $f$ has discontinuities at $t_i$, then I should take \begin{equation*} f(t_i) = \frac{f^+(t_i)+f^-(t_i)}{2} \end{equation*} where $f^+(t_i)$ and $f^-(t_i)$ are the one-sided limits. This feeling is based on Fourier series behavior at such discontinuities. But that's it, at most a feeling.