Bolzano's Theorem

​
a
1.5
b
4.5
Bolzano's theorem states that if
f
is a continuous function in the closed interval
[a,b]
with
f(a)
and
f(b)
of opposite sign, then there is a
c
in the open interval
(a,b)
such that
f(c)=0
.
​

Details

Snapshot 1: The function is positive in the interval and therefore
f(x)>0
for all
x
in
(a,b)
.
Snapshot 2: The function is negative in the interval so
f(x)<0
for all
x
in
(a,b)
.
Snapshot 3: The function is positive for
f(a)
and negative for
f(b)
, therefore there is a
c
in
(a,b)
such that
f(c)=0
.

External Links

Bolzano's Theorem (Wolfram MathWorld)
Bolzano-Weierstrass Theorem (Wolfram MathWorld)

Permanent Citation

Julio Cesar de la Yncera
​
​"Bolzano's Theorem"​
​http://demonstrations.wolfram.com/BolzanosTheorem/​
​Wolfram Demonstrations Project​
​Published: May 2, 2008