View source for Darboux theorem

{{DEF}}


If a real-valued function has a finite derivative at each point of an interval on the real axis, then, if the derivative assumes any two values on this interval, this derivative also assumes all the intermediate values on it.



====Comments====
This is a kind of intermediate value theorem (for the derivative rather than the function). The following theorem holds for functions themselves: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030190/d0301901.png" /> is a [[Continuous mapping|continuous mapping]] between metric spaces, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030190/d0301902.png" /> is a [[Connected set|connected set]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030190/d0301903.png" /> is connected. (See, e.g., [[#References|[a1]]].)

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR></table>