Theorem of the monotonous limit

Statement for the functions

Are $\backslash ]\; has,\; \backslash ,\; B\; an\; open\; interval\; (limited\; or\; not)\; and\; an\; increasing\; function$ \backslash \; F:\; \backslash ,\; has,\; \backslash ,\; B\; \backslash \; to\; \backslash \; R$.\; Then:\; *\; the\; function\; admits\; in\; any\; point$ \backslash \; x\_0$a\; limit\; on\; the\; right\; and\; a\; limit\; on\; the\; left,\; which\; one\; notes\; respectively$ \backslash \; F\; (x\_0+)$and$ \backslash \; F\; (x\_0-)$;\; they\; check\; the\; double\; inequality$ \backslash \; F\; (x\_0-)\; \backslash \; Leq\; F\; (x\_0)\; \backslash \; Leq\; F\; (x\_0+)$$

• the function admits on the terminal of right-hand side of the interval a limit, finished or not; this limit is finished if and only if $\backslash \; F$ is raised, and in the contrary case is $\backslash \; +\; \backslash \; infty$
• the function admits on the terminal of left of the interval a limit, finished or not; this limit is finished if and only if $\backslash \; F$ is undervalued, and in the contrary case is $\backslash \; -\; \backslash \; infty$

(theorem similar for the decreasing functions; he results immediately from the precedent by replacing $\backslash \; F$ by $\backslash \; -\; F$).

Statement for the continuations

If $u=\; \backslash \; left\; (u\_n\; \backslash \; right)\; \_\; \{N\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\}$ is an increasing continuation, then:

• If the continuation is raised then it is convergent.

• If the continuation is not raised then it admits $\backslash \; +\; \backslash \; infty$ for limit.

(similar theorem for the decreasing continuations; he results immediately from the precedent by replacing $\backslash \; U$ by $\backslash \; -\; U$).

Internal bonds

• monotonous Function

External bonds

• monotonous Function

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