MATH 3150 Class Notes 03-13-2025

notereal-analysis

Extreme Value Theorem

Intermediate Value Theorem

Bolzano’s Theorem

Question

Prove that any polynomial of $\mathrm{deg}=3$ has at least one real root.

Note

Something like this will definitely be on the second midterm.

Uniform Continuity

Facts

• If $f$ is continuous on $[a,b]$, then $f$ is uniformly continuous on $[a,b]$.
• If $f$ is uniformly continuous then $f$ is continuous on any interval.