The way I understand the intermediate value theorem is this: if you have a function f that is continuous over a domain $[a,b]$ then there is a value $f(c)$, where $f(a)≤f(c)≤f(b)$, such that $a≤c≤b$.
This seems self-evident. If $f$ is continuous, then there exists an $f(c)$ such that $a≤c≤b$. But isn't this just a restatement of the fact that $f$ is continuous?
Isn't the intermediate value theorem self-evident for continuous functions?