Question

tell me the historical background of intermediate value theorem

Answer 1

Intermediate
value
theorem: Let
f be a
defined
continuous
function on
[ a , b] and let
s be a
number with
f ( a) < s < f
(b ). Then
there exists
at least one
x with f (x ) =
s
In mathematical analysis , the
intermediate value theorem states that
if a continuous function , f , with an
interval , [ a , b ], as its domain, takes
values f ( a) and f (b ) at each end of the
interval, then it also takes any value
between f ( a) and f (b ) at some point
within the interval.
This has two important corollaries:
1. If a continuous function has
values of opposite sign inside an
interval, then it has a root in that
interval ( Bolzano's theorem ). [1]
2. The image of a continuous
function over an interval is itself
an interval.