Continuity

Continuity is the mathematical way of capturing an everyday intuition — that many natural processes change smoothly, without sudden jumps: a rod expanding continuously as it is heated, an organism growing continuously, a river flowing continuously, the temperature of the air varying continuously through the day. The word itself descends from the Latin continuere, "to hang together" — and the loose picture of "a graph with no breaks, drawn without lifting the pen" is a genuinely useful mental image once continuity has been properly defined. Taken as a definition on its own, though, "no breaks in the graph" is dangerously vague and can be misleading, especially for functions that are hard to sketch reliably by hand. The rigorous route is instead to build continuity directly out of the concept of a limit already developed in Section 9.2: informally, continuity at a point means "the limit there behaves exactly as the function's own value at that point would suggest."

A physical motivation — the hot-wire thermometer. Imagine a thermometer $T$ reading the temperature $t(x)$ at each point $x$ along a wire $L$. Suppose the reading holds steady at $250°\text F$ everywhere up to a particular point $x_0$, then suddenly drops to near room temperature ($75°\text F$) exactly at $x_0$ — as though insulation had been inserted there — before jumping back to $250°\text F$ immediately beyond $x_0$ (Fig. 9.32). At $x_0$ itself, nearby points on either side all read close to $250°\text F$, yet the actual reading recorded at $x_0$ is $75°\text F$: the approach of $x$ to $x_0$ has "no bearing" on the corresponding approach of $t(x)$ to $t(x_0)$. This mismatch is exactly what it means to say a function "lacks continuity," or is discontinuous, at a point.

Everyday discontinuities. Many familiar processes are naturally discontinuous as functions of some variable: switching on a light (intensity vs. time), a vehicle collision (velocity vs. time), switching off a radio (sound intensity vs. time), a bursting balloon (radius vs. air input), a breaking string (tension vs. length), postage cost as a function of parcel weight, income-tax rate as a function of taxable income, one's age in whole completed years as a function of time, and an insurance premium as a function of age. (On closer physical inspection, the first few of these — light, collision, sound, radius, tension — are not perfectly instantaneous jumps in reality, but the postage/tax/age/premium examples genuinely do jump at exact thresholds.)

The three ways continuity can fail at a point $x_0$ (Figs. 9.33–9.35). Given a graph that is otherwise unbroken on an interval $(a,b)$ except possibly at one point $x_0$, exactly three distinct things can go wrong there:

1. $f(x_0)$ is simply not defined.
2. $\lim_{x\to x_0} f(x)$ does not exist (e.g. the two one-sided limits both exist but disagree).
3. $\lim_{x\to x_0} f(x)$ exists but does not equal $f(x_0)$.

Revisiting earlier illustrations (Illustration 9.6). Section 9.2.1's two opening illustrations can now be reread through this lens: $f(x)=x^2+3$ at $x=2$ has both one-sided limits equal to $7$, and $f(2)=7$ too — nothing goes wrong, so $f$ is continuous at $2$. But $f(x)=\dfrac{x^2-16}{x+4}$ at $x=-4$ has both one-sided limits equal to $8$, while $f(-4)$ is simply undefined — failure (1) above — so this function is discontinuous at $-4$ even though its limit exists there perfectly well.

This motivates the formal definition: …