This lesson mixes the chapter without announcing which idea each question is about. Deciding whether a question is about a limit, about continuity at a point, about one of the two theorems on a closed interval, or about the quantifier order, is part of the work.

A mechanic given a car with a fault does not first ask which chapter of the manual the fault belongs to; the symptoms have to be read first. These questions are the same: the tools are all from this chapter, and which one applies is not written on the question.

A limit is a tolerance answered by a radius, and it never looks at the value at the point. Continuity is the limit agreeing with the value. On a closed bounded interval a continuous function takes every intermediate value, attains its bounds, and is uniformly continuous.

• Sequential characterisation: two sequences with different value limits kill a limit at once.
• The combining rules make almost every written formula continuous, so a discontinuity is usually at a point where something is undefined.
• Intermediate values for existence of a root; extreme values for existence of a maximum.
• Uniform continuity is a claim about one radius for the whole domain, and on a closed bounded interval it is free.

Great work! You now know how to:

• Decide which tool a question about a function is asking for
• Move confidently between limits, continuity and the two closed-interval theorems
• Read a chain of quantifiers and say which property it states