Continuity of a Function at a Point

To see what continuity should mean at a single point $x=a$, it helps to first look at three graphs of $y=f(x)$ that each fail to be continuous there in a different way (Fig. 8.1, Fig. 8.2, Fig. 8.3, described in the notes for this section).

In Fig. 8.1, the curve has a hole at $x=a$: the curve approaches a definite height as $x$ gets close to $a$ from either side, but $f(a)$ itself is simply not defined — there is no point plotted at $x=a$ at all.

In Fig. 8.2, the curve has a break at $x=a$: it rises along one line up to (but not including) $x=a$, and then continues along a different, higher line for $x\ge a$. Unlike Fig. 8.1, $f(a)$ is defined here (it belongs to the right-hand branch), but the curve still cannot be traced through $a$ without lifting the pencil, because the two branches approach different heights.

In Fig. 8.3, $f(a)$ is defined, but the plotted point sits away from the continuous line the rest of the curve is following — the curve is heading towards one height as $x\to a$, while $f(a)$ has been given a different value that is not on that natural path. …