17 December 2015

Basic calculus theorems on continuity


A compact interval [a,b] consists of the real points a and b and all real numbers between them. Continuous functions on compact intervals are basic stuff of any calculus course. The pre-calculus notion of 'continuity' is uninterrupted motion, but this too simple. The blue function below is f(x)=x sin(1/x), with additionally f(0)=0. It is continuous all right, but no uninterrupted motion will let you pass the origin.

Infinitely more intricate than the blue one is the yellow Weierstrass's monster below, an infinite sum of cosines delicately put together to yield a continuous function which has no derivative in any point. As for 'motion', there is no velocity at any moment, and you'll never get anywhere because the arc length of the graph is infinite. Zooming won't help either, because the graph is a fractal and it looks equally intricate on whatever scale.

BTW, most calculus books are wrong in defining 'f is continuous on [a,b]' as 'f is continuous in every point of [a,b]'. If they were right, the function below would not be continuous on [1,3] because it is discontinuous in the endpoints.

The right definition is: the restriction f/[a,b] is continuous in every point of [a,b]. This means that, for any c in [a,b] and any ε>0 we have |f(x)-f(c)|<ε for all x in [a,b] close enough to c.


If g(x) is continuous on [a,b], then f(x)=g((1-x)a+xb) is continuous on [0,1]. Hence, it suffices to prove theorems for continuous functions on [0,1]. Switching back and forth between [0,1] and [a,b] gives the general result. The main theorems are due to Bolzano (intermediate values), Heine (uniform continuity) and Weierstrass (extreme values). Actually, a single principle generates all three. It has nothing to do with continuity, but everything with compactness, which allows to upgrade local properties of individual points to a global property of the whole set (the interval, in our case). Here are the relevant definitions:

and this is the main theorem:

Continuity on compact intervals

The basic forms of  Heine, Bolzano and Weierstrass are immediate applications of the general principle:

and here they are in their full splendour:


Added 25th March 2015. Actually, the most general application is the theorem below, which has nothing to do with continuity and everything with topology.