The curve in Figure 1.7.2(a) represents a temperature profile $T\left(t\right)$ for one 24-hour period, where $t$ is the number of hours past midnight. It starts at $T\=55$ when $t\=0$, reaches $t\=85$, then returns to $T\=55$ at $t\=24$.

The black line represents a span of 12 hours. The endpoints sit at temperatures that are the same for two moments in time during the 24-hour period.

If $T\left(t\right)$ represents the temperature at time $t$, then the endpoints of the black line are solutions of the equation $T\left(a\+12\right)\=T\left(a\right)$, $0\le a\le 12$, or the alternative equation

Since the actual temperature function $T\left(t\right)$ isn't known, this equation can't be solved explicitly, so an argument built around it's behavior has to be used. As part of this argument, the Intermediate Value theorem will be invoked.

Define $f\left(a\right)\=T\left(a\+12\right)-T\left(a\right)$ 

If $f\left(0\right)\=0$, then $T\left(12\right)\=T\left(0\right)$, so the two times when the temperature is equal are $t\=0\,12$.

If $f\left(0\right)\ne 0$, then $f\left(12\right)\=T\left(24\right)-T\left(12\right)\=T\left(0\right)-T\left(12\right)\=-\left(T\left(12\right)-T\left(0\right)\right)\=-f\left(0\right)$
Now $T\left(t\right)$ is continuous (because temperatures are continuous), so $f\left(a\right)$ is continuous, and $f\left(0\right)\cdot f\left(12\right)<0$ because $f\left(12\right)$ and $f\left(0\right)$ have opposite signs. Hence, the Intermediate Value theorem can be applied to $f$ and there is an $\stackrel{\&Hat;}{a}\ne 0$ for which $f\left(\stackrel{\&Hat;}{a}\right)\&equals;0\&equals;T\left(\stackrel{\&Hat;}{a}\&plus;12\right)-T\left(\stackrel{\&Hat;}{a}\right)$. Thus, the two times at which the temperatures are the same are $\stackrel{\&Hat;}{a}$ and $\stackrel{\&Hat;}{a}\&plus;12$.