Continuity, differentiability and integrability

By the end you’ll be able to

• Apply the ε–δ definitions of limit and continuity and the definition of uniform continuity (continuous on a compact set implies uniformly continuous)
• Use the intermediate value, extreme value and mean value theorems to prove existence results, fixed points and inequalities
• Relate continuity, differentiability, continuous differentiability and Lipschitz conditions, using counterexamples such as $|x|$, $x^2\sin(1/x)$ and the Dirichlet and Thomae functions