Limits and Continuity

A bond's price as a function of yield behaves smoothly — nudge the yield a tiny bit, and the price moves a correspondingly tiny bit. But some financial quantities don't behave this way: an option's payoff at expiry jumps abruptly the instant the underlying crosses the strike. Telling these two behaviors apart precisely — "smooth, predictable response" versus "sudden jump" — is exactly what limits and continuity formalize, and the distinction matters enormously for how risk and hedging behave near such points.

An analogy: watching a car approach a stop sign

If you watch a car slow down as it approaches a stop sign, you can predict with confidence where it will be at the exact moment it reaches the sign, just by watching where it's headed as it gets closer and closer — even without needing to see it actually arrive. That predicted "where it's headed" is the idea of a limit: the value a function is homing in on as the input approaches some point, whether or not you ever look at the function exactly at that point. A function is continuous at a point when this predicted value actually matches what the function does there — no surprise jump when you finally arrive.

The math, one symbol at a time

The limit of $f(x)$ as $x$ approaches $a$, written

$$\lim_{x \to a} f(x) = L,$$

means: for every degree of closeness you demand between $f(x)$ and $L$ (however small), there's a corresponding closeness of $x$ to $a$ that guarantees it — you can force $f(x)$ as close to $L$ as you like by keeping $x$ close enough to $a$, without ever needing $x$ to equal $a$ exactly. A function $f$ is continuous at $a$ when three things all line up: $f(a)$ exists, $\lim_{x\to a} f(x)$ exists, and they're equal, $\lim_{x\to a} f(x) = f(a)$ — the value the function is heading toward matches the value it actually has.

Worked example 1: a removable gap versus a genuine jump

Consider $f(x) = \frac{x^2 - 4}{x - 2}$ for $x \neq 2$. Direct substitution at $x=2$ gives $0/0$, undefined — but factoring, $f(x) = \frac{(x-2)(x+2)}{x-2} = x+2$ for $x \neq 2$, so $\lim_{x\to 2} f(x) = 2+2 = 4$. The limit exists and equals 4, even though $f(2)$ itself is undefined — this is a "removable" discontinuity, a single missing point in an otherwise smooth function. Contrast this with a call option's payoff $g(S) = \max(S-100, 0)$ near $S=100$: approaching from below gives $\lim_{S\to100^-} g(S) = 0$, approaching from above gives $\lim_{S\to100^-} g(S) = 0$ too (both sides meet at 0, so the payoff itself is continuous) — but its slope has a genuine jump there, from 0 to 1, which is a different (and financially important) kind of discontinuity, in the derivative rather than the function itself.

Worked example 2: computing a limit numerically

To check $\lim_{x\to 0} \frac{\sin x}{x}$ (a limit that shows up when discretizing continuous compounding), evaluate at successively smaller $x$: at $x=0.1$, $\sin(0.1)/0.1 \approx 0.9983$; at $x=0.01$, $\approx 0.99998$; at $x=0.001$, $\approx 0.9999998$. The values are homing in on exactly 1, even though direct substitution at $x=0$ gives the undefined $0/0$ — numerically probing points closer and closer to the target is a direct, hands-on way to estimate a limit before proving it formally.

Trace the curve near different input values and watch how the output value the curve is approaching from either side is what a limit formalizes — drag close to a point and compare the left- and right-hand behavior.

What this means in practice

Continuity is the baseline assumption behind almost every pricing formula and hedging argument in finance — a hedge that relies on smoothly adjusting a position as prices move (delta hedging) implicitly assumes the relevant quantities are continuous. Where continuity breaks — option payoffs at the strike, credit events, circuit breakers — standard smooth-calculus tools like derivatives stop applying cleanly, and risk near those points behaves qualitatively differently (see the jump-in-slope example above, which is exactly why gamma spikes near an option's strike at expiry).