A step function is right continuous with left limits What does it mean that the characteristic function $f(x)=1_{[b \le x \lt \infty]}$ is right continuous with left limits? Here $x ,b \in \mathbb{R}$.

The graph of $f(x)=\mathbf{1}\{x\geq b\}$ looks like this:

Clearly, approaching any number from the right yields the same value of $f$ meaning that $f$ is right-continuous. That $f$ has left limits just means that the limit exists and is finite when approaching any number from the left. This is also obvious from the graph.
Note also what happens if the filled dot and the hollow dot swap places. Then we're looking at the graph of $f(x)=\mathbf{1}\{x>b\}$ instead, and this is left-continuous with right limits.
