The continuity of $f$ on an interval $I$ is interpreted traditionally as...

- For any $a \in I$, the value $f(a)$ is the best estimate for values of $f(x)$ when $x \in I$ is a number
**close to** - For any $a \in I$, an object moving on a line with its position at time $x \in I$ determined by $f(x)$ approaches the position $f(a)$ at times close to $a$.

- On the graph of the function $f$ for the interval $I$, there are no holes in the graph of $f$.

This condition is also expressed:

For any $a \in I$, given any $\epsilon > 0$, there is a number $\delta >0$ with the property that if $x \in I $ and $ |a-x|<\delta$, then $|f(x) - f(a)|< \epsilon$.

A dynamic visualization of the definition is found in CCD.DLCV,
Dynamic visualization of the definitions of limit and continuity.