Proof
On the
compact interval,
the function 
is bounded from above and from below, let
and 
denote the
minimum
and the
maximum
of the function. Due to
fact,
they are both obtained. Then, in particular,
for all
![{\displaystyle {}x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fda0a4b129005203d4c9b056c7aa53a1bbbe2840)
,
and so
-
Therefore,
with some
![{\displaystyle {}d\in [m,M]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97516d73669f6a2b13d597275a309e97d6dadbfb)
.
Due to
the Intermediate value theorem,
there exists a
![{\displaystyle {}c\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e96620a8d7941d7b882a2148f8703b82848ecad)
such that
.
