Postgraduate Science

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Limits & Continuity

The ε–δ definition of a limit, and what it means for a function to be continuous.

The statement $\lim_{x \to a} f(x) = L$ means: $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to (but distinct from) $a$. Made precise:

$$\forall \varepsilon > 0 \;\exists \delta > 0 : \; 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon.$$

A function $f$ is continuous at $a$ when $\lim_{x \to a} f(x) = f(a)$ — the limit exists and equals the function value. Continuity is preserved under sums, products, compositions, and (away from zeros) quotients. Polynomials and rational functions are continuous on their domains.

The intermediate value theorem and extreme value theorem are basic consequences of continuity on compact intervals.

Interactive: ε–δ visualizer

For $f(x) = x^2$ at $a = 1$: pick $\varepsilon$; the green band shows $|f(x)-L|<\varepsilon$, and the red band shows the $\delta$ that maps into it.

Quiz

1. $\lim_{x \to a} f(x) = L$ in the ε–δ sense means:
2. Continuity at $a$ requires:
3. By the intermediate value theorem, if $f$ is continuous on $[a,b]$ with $f(a) < 0 < f(b)$:
4. The squeeze theorem says: if $g(x) \leq f(x) \leq h(x)$ near $a$ and $\lim g = \lim h = L$, then:
5. $\lim_{x \to 0} \frac{\sin x}{x}$ equals: