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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:

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

p232-233