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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 [...], then we define the left limit, \( f(x_0-) \) of \( f \) at \( x_0 \) to be:

[...]

If \( x_0 \) is an adherent point of [...], then we define the right limit, \( f(x_0+) \) of \( f \) to be:

[...]

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.