Continuity and the Intermediate Value Theorem

$f$ is continuous at $x = a$ if:

1. $f(a)$ is defined.
2. $\lim_{x \to a} f(x)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$.

• Removable (hole): limit exists but $f(a)$ is undefined or different.
• Jump: left and right limits are different.
• Infinite: limit is $\pm\infty$ (vertical asymptote).

If $f$ is continuous on $[a,b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c \in (a,b)$ such that $f(c) = N$.

Application: Root Finding

If $f(a) < 0$ and $f(b) > 0$ (continuous), there's a root between $a$ and $b$.