Editing Intermediate value theorem (section)

==Practical applications==
A similar result is the [[Borsuk–Ulam theorem]], which says that a continuous map from the <math>n</math>-sphere to Euclidean <math>n</math>-space will always map some pair of antipodal points to the same place.

{{math proof|title=Proof for 1-dimensional case| proof=Take <math>f</math> to be any continuous function on a circle. Draw a line through the center of the circle, intersecting it at two opposite points <math>A</math> and <math>B</math>. Define <math>d</math> to be <math>f(A)-f(B)</math>. If the line is rotated 180 degrees, the value {{math|−''d''}} will be obtained instead. Due to the intermediate value theorem there must be some intermediate rotation angle for which {{math|1=''d'' = 0}}, and as a consequence {{math|1=''f''(''A'') = ''f''(''B'')}} at this angle.}}

In general, for any continuous function whose domain is some closed convex {{nowrap|<math>n</math>-dimensional}} shape and any point inside the shape (not necessarily its center), there exist two antipodal points with respect to the given point whose functional value is the same.

The theorem also underpins the explanation of why rotating a wobbly table will bring it to stability (subject to certain easily  met constraints).<ref>[[Keith Devlin]] (2007) [https://web.archive.org/web/20140228044921/http://www.maa.org/external_archive/devlin/devlin_02_07.html How to stabilize a wobbly table]</ref>