Math and science::Analysis::Tao::09. Continuous functions on R

# Left and right limits

The two separate halves of a complete limit $$\lim_{x \to x_0; x \in X}f(x)$$.

Let $$X$$ be a subset of $$\mathbb{R}$$, $$f: X \to \mathbb{R}$$ be a function and $$x_0$$ be a real number.

If $$x_0$$ is an adherent point of [...], then we define the left limit, $$f(x_0-)$$ of $$f$$ at $$x_0$$ to be:

[...]

If $$x_0$$ is an adherent point of [...], then we define the right limit, $$f(x_0+)$$ of $$f$$ to be:

[...]

Note: if and only if $$x_0$$ is an adherent point of $$(-\infty, x_0)$$ then it is a limit point of $$(-\infty, x_0]$$, as adherent to a set minus the element in question is the definition of being a limit point of that set.