Derivative of differentiable function satisfies intermediate value property
Contents
Name
The result is sometimes called Darboux's theorem, and is attributed to the mathematician Darboux. However, there is another more famous theorem named after Darboux.
Statement
Endpoint one-sided version
Suppose is a function defined on an interval
. Suppose:
-
(the derivative of
) exists on the open interval
.
- The right hand derivative
exists
- The left hand derivative
exists
Suppose further that and
is a real number strictly between these two numbers. Then, there exists
such that
.
Two-sided version
Suppose is a differentiable function on an open interval
. Then, the derivative
satisfies the intermediate value property on
: for
, both in
, and any value
is strictly between
and
, there exists
such that
.
Related facts
Similar facts
- Intermediate value theorem: This states that any continuous function satisfies the intermediate value property.
Opposite facts
Facts used
Proof
Proof idea
The proof idea is to find a difference quotient that takes the desired value intermediate between and
, then use Fact (3).
Proof details for one-sided endpoint version using the mean value theorem
The two-sided version follows from the one-sided endpoint version, so we only prove the latter.
Given: is a function defined on an interval
. Suppose:
-
(the derivative of
) exists on the open interval
.
- The right hand derivative
exists
- The left hand derivative
exists
Further, and
is a real number strictly between these two numbers.
To prove: There exists such that
.
Proof: Consider the function defined on the interval
as follows. By
we denote the difference quotient. Note that we use
for the right hand derivative and
for the left hand derivative:
Note that for ,
is a difference quotient between two points in
, at least one of which is one of the endpoints
.
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | ![]() ![]() ![]() |
Fact (1) | ![]() ![]() |
Follows directly from continuity of ![]() | |
2 | ![]() ![]() |
definition of derivative as a limit of a difference quotient | [SHOW MORE] | ||
3 | ![]() ![]() |
Fact (1) | ![]() |
We take the limit by plugging ![]() ![]() | |
4 | ![]() ![]() |
definition of derivative as a limit of a difference quotient | [SHOW MORE] | ||
5 | ![]() ![]() |
Steps (1)-(4) | step-combination direct. | ||
6 | There exists ![]() ![]() ![]() ![]() |
Fact (2) | ![]() ![]() ![]() |
Step (5) | By definition, ![]() ![]() ![]() ![]() ![]() |
7 | There exists ![]() ![]() |
Fact (3) | ![]() ![]() |
Step (6) | Step-fact combination direct. |
Proof details for direct proof of one-sided version
There is a direct proof that does not involve any appeal to the mean value theorem. This proof is shorter, but relies on the extreme value theorem. Note that the previous proof that relies on the mean value theorem indirectly relies on the extreme value theorem, whereas the proof below makes a direct appeal to the extreme value theorem.
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