The Intermediate Value Theorem, II
This is part two of a translation of an article that appeared in the February 2007 issue of Pythagoras.
The completeness of R, the set of real numbers, is the key to many theorems from Mathematical Analysis. You can use it to prove that many equations have a solution.
Can we distill a general theorem from yesterday’s post about the equation cos(x)=x? Yes we can and that theorem was formulated and proven in 1817 by the Czech mathematician Bernard Bolzano in an article with the wonderful title Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege. Here is the theorem, in Bolzano’s own words:
Lehrsatz. Wenn sich zwey Functionen von x, f(x) und φ(x), entweder für alle Werthe von x, oder doch für alle, die zwischen α und β liegen, nach dem Gesetze der Stetigkeit ändern, wenn ferner f(α)<φ(α), und f(β)>φ(β) ist: so gibt es jedesmahl einen gewissen zwischen α und β liegenden Werth von x, für welchen f(x)=φ(x) wird.
(Very) freely translated: when two functions are continuous on an interval [α,β] and if in addition f(α)<φ(α) and f(β)>φ(β) then the equation f(x)=φ(x) has at least one solution between α and β.
There is one thing still undefined: what does it mean for a function to be continuous? Bolzano was probably the first to give a proper definition of that.
Nach einer richtigen Erklärung versteht man unter der Redensart, das eine Function f(x) für alle Werthe von x die inner- oder außerhalb gewisser Grenzen liegen, nach dem Gesetze der Stetigkeit sich ändre, nur so viel, daß, wenn x irgend ein solcher Werth ist, der Unterschied f(x+ω)-f(x) kleiner als jede gegebene Größe gemacht werden könne, wenn man ω so klein als man nur immer will, annehemen kan.
Translated freely: one says that a function f(x) varies for all values of x according to the laws of continuity if for each value of x the difference f(x+ω)-f(x) can be made smaller than any given magnitude by taking ω small enough.
In a more strict formulation, given by Weierstraß: for any given positive number ε there is an other positive number δ such that whenever |y-x|<δ one has |f(y)-f(x)|<ε.
Yesterday’s example dealt with f(x)=x, φ(x);=cos(x), α=0, and β=1. Both functions are indeed continuous. In both cases we can always take δ=ε, for we showed that always |cos(y)-cos(x)|≤|y-x|, so that if |y-x|<δ then |cos(y)-cos(x)|<ε.
The proof of the theorem is much like yesterday’s proof. We let A={y: α≤y≤β and f(y)if |y-x|then both |f(y)-f(x)| and |φ(y)-φ(x)| are smaller than ε (apply the definition to f and to φ and take the smaller of the two δs that you get).
Because x is the least upper bound of A there is y∈A such that x-δ
Next take z such that x0.
This shows that -2εall positive ε and that means that f(x)=φ(x).
Proving that a function is continuous is not always easy, as we saw with the cosine function: we needed a trigonometric formula and an estimate for |sin(y)|. Let us see how to show that the function g, given by g(x)=x2, is continuous. Fix x and a positive ε. How do we find a suitable positive δ?
Well, we first see what we can do with |y2-x2|, how we can relate it to |y-x|. We can factor it: |y2-x2|=|y+x|×|y-x|. That gives us |y-x| but there is also the variable factor |y+x|. Since we may take δ as small as necessary (as long as it is positive) we specify at the outset that it must not be larger than 1. Why? Because then |y-x|<δ will imply that |y-x|<1 and hence that |y+x| is less than the maximum of 2|x+1| and 2|x-1|, call that maximum M. This ensures that |y2-x2|≤M|y-x| when |y-x|<δ. Now we know how to specify δ given ε: take the minimum of 1 and ε/M;. Because then: if |y-x|<δ then |y2-x2|≤M|y-x| (because |y-x|<1) and M|y-x|
Conclusion: g(x)=x2 varies according to the laws of continuity.
Now let ψ(x)=2 (constant function), and α=0 and β=2. Then g(0)<ψ(0) and g(2)>&psi(2). So there is a solution of the equation g(x)=ψ(x) between 0 and 2. So Bolzano’s theorem shows the existence of √2 in an analytic, rather than geometric, way.
Why `Intermediate Value Theorem’?
Nowadays you will mostly see the the following theorem in many books:
Intermediate Value Theorem. If f is a continuous function on an interval [a,b] then f takes on all values between f(a) and f(b) on that interval.
It does not matter whether f(a)>f(b) or f(a)
This version follows from Bolzano’s version by comparing f with the constant function with value c.
The converse is also true: work with g=f-φ. Then g(α)<0 and g(β)>0, so there is an x between α and β such that g(x)=0, or f(x)=φ(x).
Re-reading. Now re-read this post and this post. There you will find proofs of the continuity of the functions x3 and 2x. And you will also see that we used the Intermediate Value Theorem to prove the existence of ∛2 and the logarithm.
