The continuity of $f$ at $a$ is interpreted traditionally as...

- The value $f(a)$ is the best estimate for values of $f(x)$ when $x$ is a number
**close to** - An object moving on a line with its position at time $x$ determined by $f(x)$ approaches the position $f(a)$ at times close to $a$.

- On the graph of the function $f$, the point $(a,f(a))$ fills in the hole in the graph of $f$ when considered omitting that point from the graph.

This condition is also expressed:

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

To illustrate the operation of the definition of $\lim_{x \to a^-} f(x) = f(a) $ with a mapping diagram, consider an example function $f(x)=2x-\frac12$ with $a=1$, so $f(a)=1.5$ and $\epsilon = 0.5$.

Since in the definition the values $f(x)$ will be compared with $f(a)$, we begin by visualizing $f(a)$ on the target axis of the diagram, as well as the interval $(f(a)-\epsilon , f(a) + \epsilon)$.

The choice of $\delta$, here we'll use $\delta
=0.25$, is the essential feature of the definition, being made after the
selection of $\epsilon$ and controlling the numbers $x$ to which the
function is to be applied.

So the interval determined by $\delta=0.25$ is visualized on the domain axis of the diagram with the interval $(a-\delta, a+\delta)$.

So the interval determined by $\delta=0.25$ is visualized on the domain axis of the diagram with the interval $(a-\delta, a+\delta)$.

After the choice of $\delta$ the task is to show
that any $x$ in the interval $(a-\delta, a+\delta)$, must have
$f(x) \in (L-\epsilon , L + \epsilon)$. Using the arrows to visualize a
sampling of relevant $x$ and related $f(x)$ values gives a sense of what
must hold true for $\epsilon = 0.5$.

But this quality must work not just for the chosen $\epsilon$, but this ability to find an appropriate value for $\delta >0$ must be possible for any $\epsilon>0$.

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

But this quality must work not just for the chosen $\epsilon$, but this ability to find an appropriate value for $\delta >0$ must be possible for any $\epsilon>0$.

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