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 $R$ , $f : X \to R$ 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 $f : R \to R$ be the signum function:

s g n (x) := {\begin{cases} 1, & if x > 0 \\ 0, & if x = 0 \\ - 1 & if x < 0 \end{cases}

The $s g n (x)$ is continuous at every non-zero value of $x$ . The left and right limits at 0 are:

s g n (0 -) = lim_{x \to x_{0}; x \in X \cap (- \infty, 0)} s g n (x) = lim_{x \to x_{0}; x \in X \cap (- \infty, 0)} - 1 = - 1

s g n (0 +) = lim_{x \to x_{0}; x \in X \cap (0, \infty)} s g n (x) = lim_{x \to x_{0}; x \in X \cap (0, \infty)} - 1 = 1

Source

p232-233

16.03.2020