Examples Showing When Rolle’s Theorem Does Not Apply

Title: Examples Showing When Rolle’s Theorem Does Not Apply Category: mathematics

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Rolle’s Theorem does not fail when its assumptions are satisfied. Rather, examples described as “failures” usually show that the conclusion can fail when one of the assumptions is omitted. The standard hypotheses, for an interval with $a<b$, are continuity on the closed interval, differentiability on the open interval, and equal values at the endpoints.

The examples below remove one hypothesis at a time. In each case, the function has equal endpoint values where required, but there is no interior point where its derivative is zero.

Rolle’s Theorem and its hypotheses

If $a<b$, a function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and satisfies $f(a)=f(b)$, then there is at least one $c$ in $(a,b)$ such that

$$f'(c)=0.$$

Geometrically, the theorem says that a continuous curve that begins and ends at the same height, and has no corners or other failures of differentiability inside the interval, must have a horizontal tangent somewhere along the way.

Example 1: Continuity is missing

Define a function on $[0,1]$ by

$$f(x)= \begin{cases} x, & 0\le x<1,\\ 0, & x=1. \end{cases}$$

Its endpoint values are equal: $f(0)=f(1)=0$. It is differentiable at every point of $(0,1)$, with $f'(x)=1$. But it is not continuous at $x=1$: as $x$ approaches 1 from the left, $f(x)$ approaches 1, while $f(1)=0$. Since the derivative is never zero inside the interval, the conclusion of Rolle’s Theorem does not hold.

This example shows why continuity is required on the whole closed interval, including its endpoints.

Example 2: Differentiability is missing

Take $f(x)=|x|$ on $[-1,1]$. The function is continuous on the closed interval, and

$$f(-1)=f(1)=1.$$

However, it is not differentiable at $x=0$, where its graph has a corner. For $-1<x<0$, $f'(x)=-1$; for $0<x<1$, $f'(x)=1$. Thus there is no point in $(-1,1)$ where $f'(x)=0$.

Here the endpoint values match, but differentiability on the interior is missing. Continuity alone does not guarantee a horizontal tangent.

Example 3: The endpoint values are different

Let $f(x)=x$ on $[0,1]$. This function is continuous on $[0,1]$ and differentiable on $(0,1)$, but $f(0)=0$ and $f(1)=1$. Its derivative is $f'(x)=1$ everywhere, so it never equals zero.

There is no contradiction: the condition $f(a)=f(b)$ is not satisfied. A steadily increasing function can rise from one endpoint to the other without having a horizontal tangent.

What the examples show

Condition omitted	Example	Why the conclusion fails
Continuity on $[a,b]$	$f(x)=x$ for $0\le x<1$, and $f(1)=0$	The derivative is 1 throughout $(0,1)$.
Differentiability on $(a,b)$	$f(x)=	x
Equal endpoint values	$f(x)=x$ on $[0,1]$	Its derivative is 1 throughout the interval.

These are examples where the conclusion is not guaranteed after a hypothesis is removed. They do not show that the theorem itself is false.

Common mistakes

• Checking differentiability at the endpoints: Rolle’s Theorem requires differentiability only on $(a,b)$. The function must be continuous at the endpoints, but its endpoint derivatives need not exist.
• Assuming a graph must have a horizontal tangent because its endpoints match: That conclusion also relies on continuity and interior differentiability. A jump or a corner can invalidate it.
• Treating a missing hypothesis as proof that the conclusion fails: A function may still happen to have a point where $f'(c)=0$. Removing a hypothesis means the theorem no longer guarantees that result; the examples above show cases where it does not occur.
• Confusing a derivative of zero with a derivative that does not exist: At the corner of $|x|$, the derivative is undefined. That point does not count as a point where $f'(c)=0$.

Before applying Rolle’s Theorem, check the full list of hypotheses on the specified interval. If all three hold, at least one interior point with zero derivative must exist.