Darboux's theorem

Darboux's theorem states that every derivative has the intermediate value property: between any two values it takes, it takes every value in between, even if it is discontinuous.

Intermediate value property: A function has the intermediate value property if it takes every value between any two values in its range. Darboux's theorem says every derivative has this property.
Differentiability: Differentiability at a point means a function has a finite derivative there, defined by a limit of difference quotients. The theorem applies to derivatives produced by differentiable functions.
Continuity implies the intermediate value theorem: The intermediate value theorem states that continuous functions on intervals take every value between any two function values. Unlike continuity, the intermediate value property alone does not ensure a function is continuous.
Derivative of x squared sine of reciprocal x: The derivative of x² sin(1/x), extended by value zero at zero, is discontinuous but has the intermediate value property. Its derivative oscillates without settling near zero, demonstrating permitted discontinuity.
Rolle's theorem: Rolle's theorem states that a differentiable function equal at two endpoints has a point with zero derivative between them. It supplies the critical-point argument behind the intermediate-value behavior of derivatives.
Continuity: Continuity at a point means function values approach the value at that point as the input approaches it. The theorem is striking because derivative continuity is not required.
Thomae's function: Thomae's function is discontinuous at every rational number and continuous at every irrational number. It has the intermediate value property without being a derivative, illustrating that Darboux's conclusion alone is insufficient.
Derivative of x squared sine of reciprocal x squared: The derivative of x² sin(1/x²), extended by value zero at zero, is unbounded near zero and oscillates increasingly rapidly. The example shows how a derivative can behave wildly while retaining Darboux's property.
Extreme value theorem: A continuous function on a compact interval attains both a maximum and a minimum. Extrema on a closed interval help establish that derivatives cannot skip intermediate values.
Limit of a function: A limit describes the value a function approaches as its input approaches a specified point. Limits define derivatives and determine whether a derivative is continuous.

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