Definition of Continuity (on an interval): CCD.DCI
A function $f$ is defined to be continuous on an interval $I$ provided that for all $a \in I$, $f$ if continuous at $a$, i.e., for all $a \in I$, $\lim_{x \to a} f(x) = f(a)$.
The continuity of $f$ on an interval $I$ is interpreted traditionally as...
1. 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 $a$;
2. 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 $a$.
3. On the graph of the function $f$ for the interval $I$, there are no holes in the graph of $f$.
The formal definition for continuity of $f$ on an interval $I$: For any $a \in I$, given any $\epsilon > 0$, there is a number $\delta >0$ with the property that if $x \in I$ and $a-\delta< x <a+\delta$, then $f(a)-\epsilon < f(x)< f(a) + \epsilon$.
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 $|f(x) - f(a)|< \epsilon$.
Visualizing the definition of  continuity of $f$  on an interval $I$ with mapping diagrams provides visual support to understand the information of the function values for $x$ near $a$ , i.e., $x \approx a$ for any $a \in I$. When the mapping diagram is dynamic, when moving the number $a$ through the intrval $I$, the motion of the values $f(a)$ on the target axis has no noticeable jumps.

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