- A function $f$ is continuous on an open interval $(a, b)$ if $f$ is continuous at every number in $(a, b)$. - A function $f$ is continuous on an interval $[a, b)$ if $f$ is continuous on the open interval $(a, b)$ and continuous from the right at the number $a$. - A function $f$ is continuous on an interval $(a, b]$ if $f$ is continuous on the open interval $(a, b)$ and continuous from the left at the number $b$. - A function $f$ is continuous on a closed interval $[a, b]$ if $f$ is continuous on the open interval $(a, b)$, continuous from the right at $a$, and continuous from the left at $b$. --- If a function is continuous on an interval, its graph has no holes or gaps on that interval.