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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
