§ 1. The intermediate value theorem
201
z E I by hypothesis. And since certainly inf(I) ~ z ~ sup(I) for any z E I,
the set I can only be one of the intervals with end points inf(I) and sup(I),
qed.
Corollary. Every intersection of intervals is an interval.
For if x < z < y where x and y belong to the intersection I, thus to each
of the given intervals, then z also belongs to them, hence also to Ij it remains
to apply Theorem 5.
Of course the intersection in question can very well be empty: the intervals (0,1) and (2,3) have nothing in common.
Starting from these results, one easily obtains one of the most important
properties of continuous functions:
Theorem 6. (Bolzano, 1817) Let f be a real junction, defined and continuous in an interval I. Then the image f(I) of I under f is an interval.
By Theorem 5 this reduces to establishing the following result: let u, v, w
be three real numbers such that u < w < v; suppose that there exist an x E I
such that u = f(x) and ayE I such that v = fey). Then there exists a z E I
such that w = fez).
Suppose, to fix our ideas, that x < y, and consider the set E of tEl such
that both
t~y
and
f(t) ~ w.
This set is not empty - it contains x since f(x) = u < w - and it is
majorised by y. Let z = sup (E) ~ y.
Since z is the limit of points tEE satisfying f ( t) ~ w, and since f is
continuous, we have fez) ~ w. Since w < v we have z f= y and so z < y.
For z < t < y, we have f (t) > w since otherwise one would have tEE and
so t ~ z. Since f is continuous at the point z we have fez) ~ w. Whence,
finally, fez) = w, qed.
The classical formulation of Theorem 6 is to say that if the function f
takes values < 0 and values> 0 on I, then the equation f(x) = 0 has a
root in I. This is, for example, the case, for I = JR, if f is a polynomial of
odd degree. The ratio between f(x) and its term of highest degree tends to 1
when Ixl increases indefinitely as we saw - it is quite evident - in Chap. II,
nO 8, so that for Ixl large, f(x) has the sign of its term of highest degree .
. "Geometrically obvious" conclusion:
Corollary. Every algebmic equation of odd degree with real coefficients has
a real root.
But the most important consequence of Theorem 6 is the following:
201
z E I by hypothesis. And since certainly inf(I) ~ z ~ sup(I) for any z E I,
the set I can only be one of the intervals with end points inf(I) and sup(I),
qed.
Corollary. Every intersection of intervals is an interval.
For if x < z < y where x and y belong to the intersection I, thus to each
of the given intervals, then z also belongs to them, hence also to Ij it remains
to apply Theorem 5.
Of course the intersection in question can very well be empty: the intervals (0,1) and (2,3) have nothing in common.
Starting from these results, one easily obtains one of the most important
properties of continuous functions:
Theorem 6. (Bolzano, 1817) Let f be a real junction, defined and continuous in an interval I. Then the image f(I) of I under f is an interval.
By Theorem 5 this reduces to establishing the following result: let u, v, w
be three real numbers such that u < w < v; suppose that there exist an x E I
such that u = f(x) and ayE I such that v = fey). Then there exists a z E I
such that w = fez).
Suppose, to fix our ideas, that x < y, and consider the set E of tEl such
that both
t~y
and
f(t) ~ w.
This set is not empty - it contains x since f(x) = u < w - and it is
majorised by y. Let z = sup (E) ~ y.
Since z is the limit of points tEE satisfying f ( t) ~ w, and since f is
continuous, we have fez) ~ w. Since w < v we have z f= y and so z < y.
For z < t < y, we have f (t) > w since otherwise one would have tEE and
so t ~ z. Since f is continuous at the point z we have fez) ~ w. Whence,
finally, fez) = w, qed.
The classical formulation of Theorem 6 is to say that if the function f
takes values < 0 and values> 0 on I, then the equation f(x) = 0 has a
root in I. This is, for example, the case, for I = JR, if f is a polynomial of
odd degree. The ratio between f(x) and its term of highest degree tends to 1
when Ixl increases indefinitely as we saw - it is quite evident - in Chap. II,
nO 8, so that for Ixl large, f(x) has the sign of its term of highest degree .
. "Geometrically obvious" conclusion:
Corollary. Every algebmic equation of odd degree with real coefficients has
a real root.
But the most important consequence of Theorem 6 is the following: