The temperature $T(x)$ at each point $x$ on the surface of Mars (a sphere) is a continuous function. Show that there is a point $x$ on the surface such that $T(x)=T(-x)$

(Hint: Represent the surface of Mars as $\{x\in {\mathbb{R}}^{3}:||x||=1\}$.)

Consider the function $f(x)=T(x)-T(-x)$

So.....

I consider an unit sphere is locating at the origin of a $xyz$-plane.

As $||x||=1$, I can say with $radius=1=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$

To find there is a point $T(x)=T(-x)$, we use the formula $f(x)=T(x)-T(-x)$ and show somehow $f(x)$ will equal to 0 ???

It will be a point in the upper hemisphere and another point with the exactly opposite vector (if using $ijk$ plane) on the lower hemisphere

(Hint: Represent the surface of Mars as $\{x\in {\mathbb{R}}^{3}:||x||=1\}$.)

Consider the function $f(x)=T(x)-T(-x)$

So.....

I consider an unit sphere is locating at the origin of a $xyz$-plane.

As $||x||=1$, I can say with $radius=1=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$

To find there is a point $T(x)=T(-x)$, we use the formula $f(x)=T(x)-T(-x)$ and show somehow $f(x)$ will equal to 0 ???

It will be a point in the upper hemisphere and another point with the exactly opposite vector (if using $ijk$ plane) on the lower hemisphere