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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 \( X \cap (-\infty, x_0) \), then we define the left limit, \( f(x_0-) \) of \( f \) at \( x_0 \) to be:

\[ f(x_0-) := \lim_{x \to x_0; x \in X \cap (-\infty, x_0)} f(x) \]

If \( x_0 \) is an adherent point of \( X \cap (x_0, \infty) \), then we define the right limit, \( f(x_0+) \) of \( f \) to be:

\[ f(x_0+) := \lim_{x \to x_0; x \in X \cap (x_0, \infty)} f(x) \]

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.


Shorthand notations are:

\[ \lim_{x \to x_0-}f(x) := \lim_{x \to x_0; x \in X \cap (-\infty, x_0)}f(x) \]
\[ \lim_{x \to x_0+}f(x) := \lim_{x \to x_0; x \in X \cap (x_0, \infty)}f(x) \]

Example

Let \( f : \mathbb{R} \to \mathbb{R} \) be the signum function:

\[ sgn(x) := \begin{cases} \\ 1, &\quad \text{if } x > 0 \\ 0, &\quad \text{if } x = 0 \\ -1 &\quad \text{if } x < 0 \\ \end{cases} \]

The \( sgn(x) \) is continuous at every non-zero value of \( x \). The left and right limits at 0 are:

\[ sgn(0-) = \lim_{x \to x_0; x \in X \cap (-\infty, 0)}sgn(x) = \lim_{x \to x_0; x \in X \cap (-\infty, 0)}-1 = -1 \] \[ sgn(0+) = \lim_{x \to x_0; x \in X \cap (0, \infty)}sgn(x) = \lim_{x \to x_0; x \in X \cap (0, \infty)}-1 = 1 \]


Source

p232-233