Remarks

• Textbooks that use the traditional theorem shown in the introduction of this
  article may state, but do not always prove, that the result extends to intervals that may not include endpoints and to intervals that are infinite, even though the proof in this case is fairly trivial.
  [latex][/latex]
• Corollary 2 justifies the common practice of testing the sign of the derivative at a single point of an open interval with no critical values, to determine the sign over the entire interval.  In textbooks using the traditional theorem, a justification is not always formally stated.  There is an implicit understanding that the missing piece requires noting that the derivative is continuous in the interval, and therefore, by the intermediate value theorem, cannot change sign.  With Corollary 2, there is no need to check the derivative for continuity.