Bolzano Theorem (BT)
Let, for two real a and b, a < b, a function f be continuous on a closed interval [a, b] such
that f(a) and f(b) are of opposite signs. Then there exists a number
x0
[a, b] with f(x0)=0.
Intermediate Value Theorem (IVT)
Let, for two real a and b, a < b, a function f be continuous on a closed interval [a, b] such
that f(a)<f(b). Then for every y0 such that f(a)<y0<f(b) there exists a number x0[a, b]
with f(x0)=y0.
Clearly BT is only a special case of IVT. However it's interesting that the more general IVT
could be deduced from its special case, BT. Indeed, let f(a)<y0<f(b). Introduce
a new function g(x)=f(x)-y0. Then it follows that g(a)=f(a)-y0<0 whereas
g(b)=f(b)-y0>0. Therefore, g(a) and g(b) are of opposite signs. Additionally, g
is continuous wherever f is. In particular, g is continuous on [a, b] and thus satisfies
the conditions of BT. Therefore, there exists x0[a, b] such that
g(x0)=0. Written explicitly this says f(x0)-y0=0 or, finally,
Q.E.D.
A complete proof of the Bolzano Theorem appeares elsewhere.
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