intermediate value theorem

in mathematical analysis, the intermediate value theorem states that if $f$ is a continuous function whose domain contains the interval $[a,b]$ then it takes on any given value between $f(a)$ and $f(b)$ at some point within the interval

in other words, suppose the function $f(x)$ is continuous when $x$ is within the interval $[a,b]$
the intermediate value theorem says that for a constant $c$ within the interval $[a,b]$ there exists a constant $I$ where $x=I$ and $f(a)<f(I)<f(b)$

it’s a simple theorem - if your thermometer went from $30°$ to $10°$ during the sunset then there has to be a time during the sunset when your thermometer is at $20°$