Continuous and Discontinuous Functions

The word "continuity" in everyday language means an unbroken, consistent existence over a stretch of time or space: an unbroken road joining two cities, the steady flow of a river, an unbroken length of railway track, or the way a city's temperature changes gradually through the day rather than jumping about. For instance, if Pune's temperature climbs steadily from $14^{\circ}C$ at night to $29^{\circ}C$ in the afternoon, every value between $14$ and $29$ is passed through smoothly over those twelve hours — nothing is skipped. An activity that unfolds gradually, without interruption or sudden change, is called a continuous process, and its graph has no jumps, breaks, gaps or holes anywhere.

This chapter turns that everyday picture into precise mathematics for a real-valued function $y=f(x)$. Before writing down a formal definition, it is useful to look at graphs of functions that fail to be continuous, because the different ways they fail are exactly the conditions the formal definition has to rule out (covered in 8.1.1–8.1.2). The chapter then studies one-sided continuity (8.1.3), lists families of functions that are always continuous (8.1.4–8.1.5), classifies the ways continuity can break down — jump, removable and infinite discontinuities (8.1.6–8.1.9) — extends the idea from a single point to a whole interval (8.1.10), and finally uses continuity to prove that equations have roots, via the Intermediate Value Theorem (8.1.11).