Continuity

Functions without jumps, and the theorems that guarantee roots, maxima and minima — the existence results behind bisection, optimization and interpolation.

7 topics

Topics

Continuity

$f$ is continuous at $a$ if $\lim_{x\to a} f(x) = f(a)$ : small changes in the input produce small changes in the output. Continuous on an interval means you can draw the graph without lifting the pen.

Fundamental

Discontinuities

Points where continuity fails: removable (the limit exists but disagrees with the value), jump (different one-sided limits) and essential (a one-sided limit does not exist).

Fundamental

Uniform continuity

Continuity where one $\delta$ works for every point at once. On a closed bounded interval every continuous function is uniformly continuous (Heine–Cantor), which makes approximation results such as Weierstrass's theorem possible.

University

Lipschitz continuity

$|f(x) - f(y)| \le L\,|x - y|$ : the function never changes faster than rate $L$ . When the gradient is $L$ -Lipschitz, gradient descent with step $\eta < 2/L$ is guaranteed to decrease the loss.

Advanced

Bolzano's theorem

If $f$ is continuous on $[a, b]$ and $f(a)$ , $f(b)$ have opposite signs, then $f$ has a root in $(a, b)$ . It is the guarantee behind the bisection method.