A derivative need not be continuous. That is worth saying slowly, because every derivative met so far has been, and because the natural guess — that differentiating a differentiable function gives a continuous one — is false. What is true is stranger: a derivative cannot have a jump, because it always has the intermediate value property, whether or not it is continuous. So the ways a derivative can misbehave are limited, and none of them is a jump.

A speedometer that never reads a value between twenty and thirty would have to jump between them, which no speed can do. But a speed can still oscillate violently, reading every value in a range infinitely often over any interval however short — which is not continuous behaviour and breaks nothing.

The function $f(x)=x^2\sin\tfrac1x$ for $x\ne0$ with $f(0)=0$ is differentiable everywhere, and $f'$ is not continuous at $0$: the derivative there is $0$, while near $0$ the derivative oscillates between values close to $-1$ and $1$. Darboux's theorem nonetheless says that $f'$ takes every value between any two of its values, exactly as a continuous function does.

• A derivative has no jump discontinuity, by Darboux. So a function with a jump is the derivative of nothing.
• A derivative may still be discontinuous, and the oscillating example shows it; the discontinuity is essential rather than a jump.
• A function whose derivative is continuous is called continuously differentiable, and it is a strictly stronger condition than differentiability.
• Darboux is proved from the interior extremum result and not from the Intermediate Value Theorem, which needs continuity that is not available.
• At $0$ the oscillating example has $f'(0)=0$, computed from the quotient $x\sin$, which the squeeze sends to zero.

Subtract the level from the derivative by subtracting a linear function from the original, so that the question becomes finding a point where the new derivative is zero. The new function has derivative of one sign at one end and the other sign at the other, so at each end it is moving away from the end value into the interval. Its smallest value is therefore not at either endpoint, and the Extreme Value Theorem puts it at an interior point, where the interior extremum result makes the derivative vanish.

Subtract the level, so that the wanted point becomes a zero of the new derivative.

The hypothesis says the new derivative has opposite signs at the two ends.

So near each end the values inside are below the end value.

The Extreme Value Theorem applies and the minimum cannot be at either end.

The interior extremum result makes the derivative vanish there.

A derivative has the intermediate value property, whether or not it is continuous.

• A derivative need not be continuous.
• A derivative always has the intermediate value property, by Darboux.
• So a function with a jump is the derivative of nothing.
• Continuously differentiable is strictly stronger than differentiable.

Great work! You now know how to:

• Give a differentiable function whose derivative is not continuous
• State Darboux's theorem and say what it forbids
• Show that a given function is the derivative of nothing
• Say why continuously differentiable is named separately