4.11 Recap and reflection

Here we began to explore real functions $f\colon E\to\mathbb{R}$ for $E\subseteq\mathbb{R}$. Building on our knowledge of familiar functions such as polynomials and trigonometric functions, we created many new examples which exhibit interesting and sometimes counterintuitive properties.

The key definition from this section was that of a continuous function. This makes precise the intuitive idea of a function with an ‘unbroken’ graph. Arguing from the definition, we showed many familiar functions are continuous. Equally importantly, we also studied examples of discontinuous functions. We introduced the notation of a limit of a function (and related concepts such as one-sided limits) and used this to study the different ways a function can be discontinuous. We also proved various laws about adding, multiplying, forming ratios, composing and inverting continuous functions.

We observed that continuous functions $f\colon[a,b]\to\mathbb{R}$ defined over a closed, bounded interval have certain special properties. In particular, we proved three important theorems: the intermediate value theorem, the boundedness theorem and the extreme value theorem. These are all deep and powerful results, which rely on the completeness axiom. .