Darboux's theorem #
In this file we prove that the derivative of a differentiable function on an interval takes all intermediate values. The proof is based on the Wikipedia page about this theorem.
Darboux's theorem: if a ≤ b
and f' a < m < f' b
, then f' c = m
for some
c ∈ (a, b)
.
Darboux's theorem: if a ≤ b
and f' b < m < f' a
, then f' c = m
for some c ∈ (a, b)
.
Darboux's theorem: the image of a Set.OrdConnected
set under f'
is a Set.OrdConnected
set, HasDerivWithinAt
version.
Darboux's theorem: the image of a Set.OrdConnected
set under f'
is a Set.OrdConnected
set, derivWithin
version.
Darboux's theorem: the image of a Set.OrdConnected
set under f'
is a Set.OrdConnected
set, deriv
version.
Darboux's theorem: the image of a convex set under f'
is a convex set,
derivWithin
version.
Darboux's theorem: if a ≤ b
and f' a ≤ m ≤ f' b
, then f' c = m
for some
c ∈ [a, b]
.
Darboux's theorem: if a ≤ b
and f' b ≤ m ≤ f' a
, then f' c = m
for some
c ∈ [a, b]
.
If the derivative of a function is never equal to m
, then either
it is always greater than m
, or it is always less than m
.