Bolzano ‘s theorem is a theorem about continuous functions defined over an interval, which states that if a function f (x) is continuous in [a, b] and f (a) and f (b) are of different sign, it exists therefore minus a point between a and b for which f (c) = 0.
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- 1 Geometric interpretation
- 2 Demonstration
- 3 Application
- 1 Example
- 4 Bibliography
- 5 Sources
Geometrically, the theorem states that if two points (a, f (a)) and (b, f (b)) of the graph of a continuous function are located on different sides of the x-axis, then the graph intersects the axis at al minus one point between a and b.
The theorem as such does not specify the number of points, it only states that at least one exists.
Suppose that f (a) <0 and f (b)> 0. (The proof would be analogous if we assumed f (a)> 0 and f (b) <0.)
Consider the midpoint of [a, b]: (a + b) / 2.
If f ((a + b) / 2) = 0 the theorem is proved. Otherwise, f will be positive or negative at (a + b) / 2.
Take one of the halves of the interval [a, b] where the function is negative at one end and positive at the other. Let’s call the ends of this interval a1 and b1. Now let’s divide [a1, b1] in half. If f is not zero at the midpoint, it will be positive or negative. Let’s take the half where f has a different sign at each end, and call these points a2 and b2.
If we continue in this way, we obtain a succession of intervals
[a, b], [a1, b1], [a2, b2], etc., such that a <= a1 <= a2 <= … <= an and b > = b1> = b2> = …
That is, 1) The ai form an increasing sequence and the bi form a decreasing sequence.
2) Ai are always less than bi.
Let’s see what is the lemon -> + inf bn – an.
bn – an is the length of the interval [an, bn].
The length of the interval [a1, b1] is (b – a) / 2, half the length of [a, b] which is b – a.
The length of the interval [a2, b2] is (b – a) / 22, half the length of [a1, b1] which is (b – a) / 2.
And continuing in this way, the length of the interval [an, bn] is (b – a) / 2n.
3) limn -> + inf bn – an = limn -> + inf (b – a) / 2n = 0.
1), 2) and 3) are the conditions of the definition of PSMC:
an is increasing , bn is decreasing
For all natural n an <bn
For all ε> 0 there is natural h / bh – ah <ε (which is the same as limn -> + inf bn – an = 0.)
Every PSMC has the property of defining a boundary number between both sequences.
((an), (bn)) is a PSMC => there is c / for all n an <= c <= bn, lim an = c- and lim bn = c +.
lim an = c- means that for all δ> 0 there is n1 / for all n> = n1 c – δ <an <c.
lim bn = c + means that for all δ> 0 there is n2 / for all n> = n2 c <bn <c + δ.
So taking the largest between n1 and n2, let’s call it n3, both are true.
That is, for all δ> 0 there is n3 / for all n> = n3 c-δ <[an, bn] <c + δ.
This means that, for any c environment we consider, there is an interval [an, bn] contained in that environment.
On the other hand, f is continuous in [a, b] by hypothesis. Therefore it is continuous in c. By definition of continuity, limx-> cf (x) = f (c).
If f (c) <0, by theo. For conservation of the sign there exists an environment of c where f (x) is negative. Within this environment, there exists an interval [an, bn], where f (an) is of a different sign than f (bn). This is a contradiction, therefore f (c) cannot be negative. If f (c)> 0, by conservation of the sign there exists an environment of c where f (x) is positive. But, again, within that environment there exists an interval [an, bn] such that f (an) is of a different sign than f (bn). Therefore, there is no other possibility: f (c) = 0.
Bolzano’s theorem allows the location of the roots of a continuous function by applying the bisection method, which is a numerical calculation method, for which it divides into two subintervals.
If the function changes sign over an interval, the value of the function is evaluated at the midpoint c = (a + b) /2. If if “c” equals zero, it is the sought root. Otherwise, the sign of f (c) is analyzed to see if it is opposite with f (a) or with f (b). The interval [a, c] or [c, b] in which a change of sign occurs is taken. The process is repeated successively for a smaller and smaller interval, until the desired value is found or approximated.
Check that the equation x 3 + x – 1 = 0 has at least one real solution in the interval [0,1].
We consider the function f (x) = x 3 + x – 1, which is continuous in [0,1] because it is polynomial. We study the sign at the ends of the interval:
f (0) = −1 <0
f (1) = 1> 0
As the signs are different, the Bolzano theorem is true, therefore there is a c in the open innervale (0; 1) such that f (c) = 0. This shows that there is a real root in this open interval.