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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.