Left and right limits and discontinuities

Let $X \subseteq R$ and let $x_{0} \in X$ . Let $f : X \to R$ be a function.

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):

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

These are discussed further on the back side.

16.03.2020