Continuity and the ε–δ Definition

A function $f$ is continuous at $a$ if we can force $f(x)$ as close to $f(a)$ as we like by keeping $x$ close enough to $a$. Precisely, the $\varepsilon$–$\delta$ definition:

Again a challenge–response game: the adversary sets an output tolerance $\varepsilon$; you must supply an input window $\delta$ that keeps the output inside $\varepsilon$.

Worked example. Show $f(x) = 3x + 1$ is continuous at every $a$.

Given $\varepsilon > 0$, notice $|f(x) - f(a)| = |3x - 3a| = 3\,|x - a|$. Choose $\delta = \dfrac{\varepsilon}{3}$. Then $|x-a| < \delta$ gives $|f(x)-f(a)| = 3|x-a| < 3\delta = \varepsilon$. Done — one clean choice of $\delta$ works for every $a$.