4. Continuity

Continuity via ε-δ, uniform continuity, and the great theorems: the intermediate and extreme value theorems.

• The ε–δ Definition — This scene makes the limit definition visible. It stacks an ε-band over a δ-band and grid-checks that the promise really holds. The order is the point. First, someone else picks the allowed error. Then you must find a matching input window. If the input stays in that window, the output stays in the error band. The formula below states the promise. For the line here, a window of half the error is enough. For example, play at throwing a ⚽ ball into a 📦 box. Your friend may swap in a smaller and smaller box. Standing close enough, you can still make every throw. The smaller the box, the closer you must stand.
• The Definition of Continuity — This scene shows what continuity means at a point. Move the point a, then lift one value and watch continuity break. There is one condition. The height you approach must equal the height at the point. Changing a single value breaks that agreement. The graph then tears at that point. For example, draw the graph with a ✏️ pencil without lifting it. If one stroke suffices, the function is continuous. If one point floats above the rest, you must lift the pencil there.
• Thomae's Function — This scene shows Thomae's function, a popcorn of dots. Dots jump up at rationals and stick to the floor at irrationals. Tall dots are rare. Fix any height: only finitely many dots reach it. So near an irrational you can choose a window avoiding tall dots. That is why it is continuous at irrationals only. For example, picture a jar of 🍬 candies. Only a few are big, and you can count them. A corner with only tiny candies is nearly flat. Neighborhoods of irrationals look like that: no big jumps, so the function stays quiet.
• Three Faces of Continuity — This scene checks continuity in three different ways. The ε–δ argument, the sequential test, and preimages of open sets stand side by side. The three faces say the same thing. Bands, sequences, or pulled-back open sets all give the same verdict. A smooth curve passes all three. A jump function fails all three at the same spot. For example, look at one object in a 🪞 mirror from three sides. Front or side, it is the same object. Continuity is like that: many checks, one question.
• Discontinuities of Monotone Functions — This scene inspects the discontinuities of a monotone function. It only ever climbs, so every break is a jump. Big jumps are few. The total climb is fixed, and that bounds them. Set a threshold: only finitely many jumps exceed it. Lowering the threshold step by step counts all discontinuities. For example, take a 🪜 staircase three meters tall in total. Steps higher than 30 cm number at most nine. Counting steps down to 3 cm changes nothing: many steps, still countable one by one.
• Uniform Continuity — This scene tests uniform continuity. Slide the point x and watch how the δ it needs changes from point to point. Ordinary continuity lets δ depend on the point. Uniform continuity demands one δ working everywhere. Where the curve steepens, the needed δ keeps shrinking. If it steepens forever, no single δ can cover everything. For example, imagine a whole class sharing one 📏 ruler. Everyone must measure with that same ruler. If it is too short for even one person's turn, the measurement fails there.
• You Can't Climb Without Crossing Every Height — This scene checks the intermediate value theorem. Move the target height and find where the graph crosses it. An unbroken path cannot skip the middle. Going from low to high, it must pass every height between. If the graph breaks, the promise fails. For example, watch the column of a 🌡️ thermometer. Rising from 10° in the morning to 20° at noon, it must pass 15°. The column never teleports.
• The Extreme Value Theorem — This scene checks the extreme value theorem. On a closed interval the maximum is attained; open intervals and breaks show why it can fail. Two conditions are needed. The interval must be closed, and the graph must be unbroken. Drop either one, and the highest point may only be approached, never reached. That is why both hypotheses matter. For example, picture a 🧑 hiker walking a trail to the summit. If the trail runs all the way, some step lands on the highest point. If the trail breaks just before it, the hiker gets close but never arrives.
• When the Limit Breaks — This scene separates pointwise from uniform convergence. Watch the power curves break in the limit. Pointwise convergence waits at each point separately. Uniform convergence moves the whole interval at once. If even one point holds out forever, convergence is not uniform. That is how a limit of continuous curves can break. For example, picture 🧑 classmates in a row sitting down together. It counts as done only when everyone is seated. If one classmate stays standing, the row as a whole is not finished.