Continuity

A continuous function is simply one whose graph has no breaks, so you can draw it without lifting your pencil. It matters because continuous functions behave predictably: small changes in the input give small changes in the output, and plugging in gives the limit. The definition $\lim_{x\to a} f(x) = f(a)$ is just two things being equal: $\lim_{x\to a} f(x)$ (where the function is heading as you approach $a$) and $f(a)$ (where it actually is when you get there). The Intermediate Value Theorem looks long, but it says something you already believe: if a pencil line starts below some height and ends above it, it crossed that height somewhere. Bear with the symbols; the ideas are simple.

Every lesson so far has made a point of separating where $f$ is heading from what $f(a)$ actually is. Continuity is the happy case where those two match. That takes three things: $f(a)$ is defined, $\lim_{x\to a} f(x)$ exists, and they're equal.

Kinds of discontinuity. Each one is a way those three can fail, and you've met all of them already:

• Removable: a hole, like $\frac{x^2 - 1}{x - 1}$ at $1$. The limit exists but $f(a)$ is missing or wrong. Redefining that one point fixes it.
• Jump: the one-sided limits differ.
• Infinite: a vertical asymptote.

Continuous everywhere on their domains: polynomials, rational functions, roots, trig functions, $e^x$, $\ln x$, and their sums, products, quotients and compositions. This is really the limit laws in disguise: they told us the limit of a polynomial is its value, and that's exactly what continuous means. It's also why plugging in has worked so often.

The Intermediate Value Theorem is where continuity starts paying off: it guarantees solutions exist. Suppose $f$ is continuous, $f(1) < 0$ and $f(2) > 0$. The graph starts below the $x$-axis and ends above it, and since it has no jumps, it can't get from one side to the other without crossing. So $f(c) = 0$ for some $c$ between 1 and 2. The theorem doesn't say where, but you can hunt the root down by checking the midpoint and keeping whichever half still has a sign change. That's the bisection method.

The IVT is why bisection root-finders never fail on a sign change, and continuity is the formal version of "small changes in input cause small changes in output," which every stable control system depends on.