antipodal isothermic points

Assume that the momentary temperature on any great circle of a sphere varies continuously (http://planetmath.org/Continuous).  Then there exist two diametral points (i.e. antipodal points, end points of a certain diametre (http://planetmath.org/Diameter)) having the same temperature.

Proof.  Denote by $x$ the distance of any point $P$ measured in a certain direction along the great circle from a and let $T(x)$ be the temperature in $P$.  Then we have a continuous (and periodic (http://planetmath.org/PeriodicFunctions)) real function $T$ defined for  $x\geqq 0$  satisfying  $T(x+p)=T(x)$  where$p$ is the perimetre of the circle.  Then also the function $f$ defined by

$$f(x):=T\left(x+\frac{p}{2}\right)-T(x),$$

i.e. the temperature difference in two antipodic (diametral) points of the great circle, is continuous.  We have

$f\left({\displaystyle \frac{p}{2}}\right)=T(p)-T\left({\displaystyle \frac{p}{2}}\right)=T(0)-T\left({\displaystyle \frac{p}{2}}\right)=-f(0).$(1)

If $f$ happens to vanish in  $x=0$,  then the temperature is the same in  $x=\frac{p}{2}$  and the assertion proved.  But if  $f(0)\ne 0$,  then by (1), the values of $f$ in  $x=0$  and in  $x=\frac{p}{2}$  have opposite signs.  Therefore, by Bolzano’s theorem, there exists a point $\xi$ between $0$ and $\frac{p}{2}$ such that  $f(\xi )=0$.  Thus the temperatures in $\xi$ and $\xi +\frac{\pi }{2}$ are the same.

Reference:  http://www.maths.lth.se/query/Fråga Lund om matematik, 6 april 2006