3. Continuity

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

• The ε–δ Definition — The rigorous meaning of $\lim_{x\to a} f(x) = L$: no matter how narrow the $\varepsilon$-band, there is a $\delta$-band around $a$ where $f$ stays inside it. Check $\delta = \varepsilon/2$ for $f(x) = 2x$, $a = 1$, $L = 2$.
• The Definition of Continuity — $f$ is continuous at $a$ when $\lim_{x\to a} f(x) = f(a)$ — the limit and the value must agree. Move the point $a$; in broken mode the value at one point is lifted and the definition breaks.
• Discontinuities of Monotone Functions — A monotone function has left and right limits everywhere, so every discontinuity is a jump. Only finitely many jumps can exceed $1/n$, so the discontinuities are at most countable — no matter how many there are.
• Uniform Continuity — Uniform continuity demands one $\delta$ that works at every point $x$ — $\delta$ may depend on $\varepsilon$ alone, never on $x$. Watch the required $\delta$ shrink as $x$ grows for $f(x) = x^2$, so no single $\delta$ covers all of $\mathbb{R}$.
• You Can't Climb Without Crossing Every Height — A continuous function on $[a,b]$ must step on every value between $f(a)$ and $f(b)$. Move the target level $c$ and find the crossing $\xi$.
• The Extreme Value Theorem — A continuous function on a closed interval always attains its max and min. Verify it for $f(x) = x^3 - 3x$ on $[-2,2]$, and see from the open-interval and discontinuous counterexamples that both hypotheses are necessary.
• When the Limit Breaks — $f_n(x)=x^n$ converges at every point of $[0,1]$, yet $\sup|f_n-f|=1\not\to 0$: not uniformly. Watch the limit break at $x=1$.