Is a function like $f(x) = \frac{2^x - 1}{x}$ continuous? Its plot shows that it's continuous, but the function doesn't have a value at $x=0$, is that right? If not, is it still a continuous function?
The function $$f(x) = \frac{2^x - 1}{x}$$ is continuous in all the domain of definition that is $x\in \mathbb{R}\setminus\{0\}$.

Since for $x=0$ the limit exists finite we can define $$f(0)=\lim_{x\to 0} f(x)=\ln 2$$ and by this extension $f(x)$ becomes continuous for all $x\in\mathbb{R}$.
$f(0)$ is the derivative of $2^x$ at $0$. You can extend it continuously at $0$.