What is Continuity?

Example 21.17

The equation $x^3 + x = 1$ has a unique real solution. Indeed $f(x) = x^3 + x$ is continuous and strictly increasing on $\R$ (as a sum of strictly increasing functions), $\lim_{-\infty} f = -\infty$ and $\lim_{+\infty} f = +\infty$, so the bijection theorem applies with $k = 1$. Since $f(0.6) = 0.816 < 1$ and $f(0.7) = 1.043 > 1$, the solution lies in $\intoo{0.6}{0.7}$.

Example 13.7 (A taxonomy of discontinuities)

Three ways to fail at a point, in increasing severity. Removable: $f(x) = \frac{\sin x}{x}$ on $\R^*$ has limit $1$ at $0$; defining $f(0) = 1$ repairs it — the discontinuity was a hole, not a feature. Jump: $\lfloor x \rfloor$ at an integer has distinct one-sided limits ($n - 1$ and $n$); no choice of value can reconcile them, but both half-limits exist. Essential: $\sin\frac1x$ at $0$ has no one-sided limit at all (Exercise 13.1) — oscillation without settlement. Monotone functions can only produce the middle kind (their one-sided limits always exist, being suprema and infima), which is why their discontinuity sets are at most countable — one rational per jump. Derivatives, by Darboux’s theorem (Exercise 14.10), can only produce the last kind: a function with a jump discontinuity is never the derivative of anything.

Example 13.9 (Max and min of continuous functions)

If $f, g$ are continuous, so are $\max(f, g)$ and $\min(f, g)$: no case analysis needed, thanks to the identities

and the continuity of sums and of $\abs{\,\cdot\,}$ (Proposition 13.8). In particular $f^+ = \max(f, 0)$ and $f^- = \max(-f, 0)$ are continuous with $f = f^+ - f^-$: the sign-splitting used for series (Chapter 17) and, at full scale, in the integration theory of the Year 3 volume, costs nothing in regularity.

Example 13.11 (One equation, the full protocol)

Solve $\eu^x = 3 - x$ over $\R$: existence, uniqueness, location. Set $g(x) = \eu^x + x - 3$, continuous. Location and existence: $g(0) = -2 < 0$ and $g(1) = \eu - 2 > 0$, so the intermediate value theorem plants a solution in $\intoo{0}{1}$. Uniqueness: $g$ is a sum of the strictly increasing $\eu^x$ and $x - 3$, hence strictly increasing on $\R$; a strictly monotone function takes each value at most once, so the solution is unique on all of $\R$ (not just in the interval probed). The protocol — rearrange to $g = 0$, sign change for existence, monotonicity for uniqueness — settles most “how many solutions” questions in three lines, and Chapter 14’s variation tables extend it to non-monotone $g$ by cutting $\R$ into monotone branches.