let $f:[a,b] \rightarrow \mathbb{R}$ such that $f$ is continuous on $[a,b]$.
Is the negation of this the following: $\exists l\in[a,b]:\lim_{x\to l}f(x) \neq f(l)$.
If so, does this mean that the $\lim_{x\to l}f(x)$ exists but is not equal to $f(l)$ or does it mean that the limit can either exist or not exist, and if it does exist, it is not equal to $f(l)$
Yes, and it means that either the limit $\lim_{x\to l}f(x)$ doesn't exist or, if it exists, it is not equal to $f(l)$.