Editing Borsuk–Ulam theorem (section)

===1-dimensional case===
The 1-dimensional case can easily be proved using the [[intermediate value theorem]] (IVT).

Let <math>g</math> be the odd real-valued continuous function on a circle defined by <math>g(x)=f(x)-f(-x)</math>. Pick an arbitrary <math>x</math>. If <math>g(x)=0</math> then we are done. Otherwise, without loss of generality, <math>g(x)>0.</math> But <math>g(-x)<0.</math> Hence, by the IVT, there is a point <math>y</math> at which <math>g(y)=0</math>.