The Intermediate Value Property of Derivatives

Optimization

The Intermediate Value Theorem applies to continuous functions. Darboux's Theorem shows that derivatives satisfy an analogous property, a derivative takes every value between its values at the endpoints of an interval, even if the derivative itself is not continuous.

Background

The Intermediate Value Theorem states: if f is continuous on [ a , b ] , then f takes every value between f ( a ) and f ( b ) .

In this section we prove that differentiable functions share a similar property through their derivatives. If f is differentiable on [ a , b ] , then f' takes every value between f'(a) and f'(b), even though f' need not be continuous.

Darboux's Theorem

Darboux's Theorem. If f is differentiable on the interval [ a , b ] , and if k is any value between f'(a) and f'(b), then there is at least one point c ∈ ( a , b ) such that f'(c)=k.

Proof Suppose f'(a). (The case f'(a)>k>f'(b) is analogous.) Define the auxiliary function g ( x ) = k x − f ( x ) on [ a , b ] . Since f is differentiable (and hence continuous) on [ a , b ] , g is also differentiable and continuous on [ a , b ] . By the Extreme Value Theorem, g attains its absolute maximum on [ a , b ] . Since g'(a)=k-f'(a)>0, it follows from the definition of the derivative that g ( x ) > g ( a ) for every x ∈ ( a , b ) sufficiently close to a . Therefore the absolute maximum of g does not occur at x = a . Since g'(b)=k-f'(b)<0, similarly g ( x ) > g ( b ) for every x ∈ ( a , b ) sufficiently close to b . Therefore the absolute maximum of g does not occur at x = b either. Hence g attains its absolute maximum at some interior point c ∈ ( a , b ) . By Fermat's Theorem, g'(c)=0, i.e., k-f'(c)=0\Rightarrow f'(c)=k.

Significance

Darboux's Theorem is remarkable because it gives derivatives a property, the intermediate value property, that we normally associate only with continuous functions, yet it requires no continuity assumption on f' whatsoever. In particular:

  • It is impossible for f' to "skip" a value. If f'(a)=1 and f'(b)=3, then f' must equal $2$ somewhere in ( a , b ) .
  • A derivative cannot have a simple "jump discontinuity" (a discontinuity where the one-sided limits exist but differ). Any discontinuity of f' must be of a more exotic type (such as an oscillating discontinuity).

Frequently Asked Questions

Is Darboux's Theorem the same as the Intermediate Value Theorem for derivatives? Yes, informally. Darboux's Theorem plays exactly the role for derivatives that the Intermediate Value Theorem plays for continuous functions. The difference is that Darboux's Theorem requires f to be differentiable (not merely that f' is continuous), while the IVT requires f itself to be continuous.

Can a derivative be discontinuous? Yes. A classic example is f ( x ) = x 2 sin ⁡ ( 1 / x ) for x ≠ 0 and f ( 0 ) = 0 . One can show that f'(0)=0 but f'(x)=2x\sin(1/x)-\cos(1/x) oscillates near x = 0 and has no limit there. So f' is defined everywhere but is discontinuous at $0$. Darboux's Theorem tells us that despite this discontinuity, f' still satisfies the intermediate value property.

Does Darboux's Theorem imply that f' is continuous? No. The intermediate value property is weaker than continuity. A function can satisfy the intermediate value property without being continuous. Darboux's Theorem guarantees the former for every derivative, but derivatives can be discontinuous (as the example above shows).
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