This is a question I understand intuitively but am having trouble proving rigorously:
Let $f$ have a jump discontinuity at $x_0$. Show that if $x_1, x_2, . . .$ is any sequence of points in the domain of $f$ converging to $x_0$, with no $x_j$ equal to $x_0$, then the sequence $f(x_1), f(x_2), . . .$ has at most two limit-points.
Since there is a jump, there is going to be a limit point from the left and from the right. I am picturing a graph where there is a break in the line, so from the left $f(x)$ will approach say $a$, and from the right $f(x)$ will approach $b$. How can I prove that there cannot be greater than two limit points?