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 $x ,b \in \mathbb{R}$.

Solution 1:

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.

Solution 2:

It means that at every point $x_0\in \mathbb{R}$, both one-sided limits $$ \lim_{x \nearrow x_0} f(x) \textrm{ and } \lim_{x \searrow x_0} f(x) $$ exist, and furthermore that $f(x_0) = \lim_{x \searrow x_0} f(x)$. In this example the function is fully continuous (both one-sided limits are equal to the function) everywhere except at $b$, where it fails to be left-continuous, but still has a limit.