Temperature defined on a tetrahedron I am asked to prove that the temperature of a tetrahedron must have at least three distinct points on the edges or vertices of the tetrahedron with the same value. I may assume that the temperature is a continuous function.
Is the following reasoning correct?

Consider two vertices $a,b$ and suppose they have temperatures $T_a,T_b$. There are three distinct paths along the vertices and edges of the tetrahedron from $a$ to $b$. By the intermediate value theorem, for any $T_c \in (T_a,T_b)$, there exists at least one point on each of the three paths such that the temperature at those points is equal to $T_c$.

Your argument implicitly assumes that $T_a\ne T_b$.  (In fact, your notation $(T_a,T_b)$ also suggests that $T_a<T_b$.) This need not be the case. To complete the proof, consider two cases: 


*

*All vertices have the same temperature. [conclusion is immediate]

*There are two vertices of unequal temperature: label them $a$ and $b$ so that $T_a<T_b$. [proceed as above]

