Here is an example of the Intermediate Value Theorem that is more rely on real life. We claim that for every circle in the universe, there exist antipodal points of the circle that has the same temperature. We let the function be f(θ) = T(θ) −T(θ+π) which T(θ) is the temperature of any point P_θ on the circle and T(θ+π) is the temperature of the point P_θ+π, which is the antipodal point of P_θ. So if f(x) equals to zero, that means T(θ) is equal to T(θ+π), which satisfy our claim.
In order to prove our claim, we can use the Intermediate Value Theorem, which states that for an interval [l,r] of a continuous function, we are able to find a point c such that f(c)=L, which is between f(l) and f(r). Imagining we are flying on a plane from the Southern Hemisphere to the Northern Hemisphere, no matter where are we taking off, we will fly across the equator for sure. Coming back to our example, when we start at the x-axis, which θ = 0, we may also get f(0) < 0. On the other hand, when we start at θ = π, we may get f(π) > 0. Since there is a temperature for every point on the circle, we are able to find a point P_a such that f(a) = 0 and f(a) equals to f(a+π). Finally, this prove rely explicitly on the concept of continuity because it is told that there is a temperature on each point.