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

Left and right limits and discontinuities

Let $$X \subseteq \mathbb{R}$$ and let $$x_0 \in X$$. Let $$f: X \to \mathbb{R}$$ be a function.

If $$f(x_0-)$$ and $$f(x_0+)$$ both exist and are both equal to $$f(x_0)$$, then $$f$$ is continuous at $$x_0$$.

$$x_0$$ is an element of $$X$$ by definition, so $$f(x_0)$$ must exist and be defined as $$f$$'s range includes all of $$X$$.

$$x_0$$ must be an adherent point of both $$X \cap (-\infty, x_0)$$ and $$X \cap (x_0, \infty)$$, as these are requirements for the left and right limits to exist.

Discontinuities

Note that we require both ① limits to exist and ② be equal to each other and ③ to $$f(x_0)$$, and ④ that the statement is only an implication, not a bi-implication. Discontinuities can be found by looking at these properties. Four discontinuities that arise at (at least):

1. [...] discontinuity and [...] discontinuity
2. [...] discontinuity
3. [...] discontinuity
4. [...] discontinuity, with function continuity maintained (I made up this name). Not 100% if it exists.

These are discussed further on the back side.