Editing Derivative test (section)

===Precise statement of monotonicity properties===
Stated precisely, suppose that ''f'' is a [[real number|real]]-valued function defined on some [[open interval]] containing the point ''x'' and suppose further that ''f'' is [[Continuous function|continuous]] at ''x''.

* If there exists a positive number ''r''&nbsp;>&thinsp;0 such that ''f'' is weakly increasing on {{open-closed|''x'' − ''r'', ''x''}} and weakly decreasing on {{closed-open|''x'', ''x'' + ''r''}}, then ''f'' has a local maximum at ''x''.
* If there exists a positive number ''r''&nbsp;>&thinsp;0 such that ''f'' is strictly increasing on {{open-closed|''x'' − ''r'', ''x''}} and strictly increasing on {{closed-open|''x'', ''x'' + ''r''}}, then ''f'' is strictly increasing on {{open-open|''x'' − ''r'', ''x'' + ''r''}} and does not have a local maximum or minimum at ''x''.
Note that in the first case, ''f'' is not required to be strictly increasing or strictly decreasing to the left or right of ''x'', while in the last case, ''f'' is required to be strictly increasing or strictly decreasing. The reason is that in the definition of local maximum and minimum, the inequality is not required to be strict: e.g. every value of a [[constant function]] is considered both a local maximum and a local minimum.