False belief: A function being continuous in some open interval implies that it is continuous on some point in that interval:
Counterexample:
The Weierstrass function is an example of a function that is continuous everywhere but differentiable nowhere:
$f(x) = \sum_{n=0}^\infty a^n \cos(b^n \pi x)$
Where $a \in (0, 1)$, $b$ is a positive odd integer, and $ab > 1 + \frac{3\pi}{2}$. The function has fractal-like behavior, which leads to it not being differentiable. This notion is rather disheartening to most calculus students, though!