Continuity

$f$ is continuous at $x_0$ if three conditions hold:

1. $f(x_0)$ is defined (the point belongs to the domain);
2. $\lim_{x\to x_0}f(x)$ exists (left and right limits coincide);
3. $\lim_{x\to x_0}f(x) = f(x_0)$ (limit and value coincide).

If any of these three conditions fails, $f$ has a discontinuity at $x_0$.

Types of discontinuity:

• Removable (first kind): the limit $L=\lim_{x\to x_0}f(x)$ exists and is finite but $f(x_0)\neq L$ (or $f(x_0)$ is undefined). The graph has a "hole" that can be "plugged" by redefining $f(x_0)=L$. Example: $f(x)=(\sin x)/x$ at $x=0$ (limit is $1$ but undefined).
• Jump (second kind): both one-sided limits exist and are finite but are different: $\lim_{x\to x_0^-}f(x) \neq \lim_{x\to x_0^+}f(x)$. The difference $|\lim^+ - \lim^-|$ is the jump size. Example: the floor function $f(x)=\lfloor x\rfloor$ has jumps of size 1 at every $x\in\mathbb{Z}$.
• Second kind: at least one one-sided limit is infinite or does not exist. Includes vertical asymptotes and violent oscillations like $f(x)=\sin(1/x)$ at $x=0$.

$f$ is continuous on an interval if it is continuous at every interior point and at the endpoints (with the appropriate one-sided limit).

Three fundamental theorems apply to functions continuous on a closed bounded interval $[a,b]$.

Bolzano's theorem (zero existence): if $f$ is continuous on $[a,b]$ and $f(a)\cdot f(b)<0$ (i.e., $f$ changes sign), then there exists at least one $c\in(a,b)$ such that $f(c)=0$. Geometric interpretation: a continuous curve that starts above the $x$-axis and ends below (or vice versa) must cross it. Useful for locating roots (bisection method).

Weierstrass's theorem (extreme value): if $f$ is continuous on the closed bounded interval $[a,b]$, then $f$ is bounded and attains its maximum and minimum values — i.e., there exist $x_m,x_M\in[a,b]$ with $f(x_m)=\min f$ and $f(x_M)=\max f$. Both hypotheses (continuity and closed bounded interval) are necessary: $f(x)=1/x$ on $(0,1]$ has no maximum because the interval is not closed.

Intermediate value theorem: if $f$ is continuous on $[a,b]$, then $f$ takes every value between $f(a)$ and $f(b)$. More precisely, for every $k$ between $f(a)$ and $f(b)$ there exists $c\in(a,b)$ with $f(c)=k$. Generalises Bolzano (which is the case $k=0$ with opposite signs).