Given a function $f : \mathbb{R} \to \mathbb{R}$, is it necessary that if $x=t$ is the first point of discontinuity, then $\exists \ y \in \mathbb{R}$ such that $f(t) = y$ would make the function continuous over the interval $(-\infty, t]$?
My hunch was to consider the sequence $x_{n \in \mathbb{N}} = f(t-\frac{\epsilon}{n})$ and say that its limit will be the desired point. But I don't know if every sub-sequence $y_n \subseteq f$ will converge to the same point.