#math/calculus **Intermediate Value Theorem.** Let $f$ be continuous on $[a,b]$. Every value strictly between $f(a)$ and $f(b)$ is attained at some point between $a$ and $b$: $ \min\{f(a),f(b)\}<c<\max\{f(a),f(b)\} \quad\Longrightarrow\quad \exists\,x_{0}\in(a,b)\text{ such that }f(x_{0})=c. $ At an endpoint, continuity only requires the appropriate one-sided limit from within $[a,b]$. **Contrapositive form.** If $f(x)\neq c$ for every $x\in[a,b]$, then $f-c$ cannot change sign on the interval. Thus $f(a)<c$ implies $f(b)<c$, and $f(a)>c$ implies $f(b)>c$. A useful special case is the **root test**: if $f$ is continuous on $[a,b]$ and $f(a)f(b)<0$, then $f(x_{0})=0$ for some $x_{0}\in(a,b)$. The theorem guarantees existence, not uniqueness. > [!Practice] > Marsden & Weinstein, p. 145, selected exercises from $\S 3.1$.