Continuity
A limit predicts where a function is heading, and the value says where it actually is. When do the two agree, and what can be proved about a function where they always do?
Where you are 5 inside
What this area is about
A graph drawn without lifting the pencil is the picture. The exact version uses limits: f is continuous at a when \lim_{x \to a} f(x) exists and equals f(a). Continuity lets a limit be found by plugging in, and it is the hypothesis behind the big existence theorems of calculus.
How these fit together
Definitions. Continuity at a point states the test above. Continuity on an interval asks for it at every interior input, adds one sided continuity at the ends of an interval such as [a, b], and lists the familiar functions that are continuous wherever they are defined.
Classifying breaks. Discontinuities and how to remove them sorts the failures into removable, jump, infinite and oscillating, using the two one sided limits, and repairs the removable ones.
Procedure. Making a piecewise function continuous chooses constants so each rule of a piecewise function meets the next at its boundary.
Result. The extreme value theorem says that a function continuous on a closed interval [a, b] takes a largest and a smallest value there.
A path through it
Continuity at a point first, then on an interval. After that, discontinuities and then piecewise functions; the extreme value theorem can come any time after continuity on an interval.
Next come the intermediate value theorem, extrema on a closed interval and differentiability.