The inequality and its applications

The mean value inequality

If $f$ is continuous on $[a \,;\, b]$, differentiable on $\left]a \,;\, b\right[$, and if $\left\lvert f'(x) \right\rvert \leq k$ for every $x$ in $\left]a \,;\, b\right[$, then

$$\left\lvert f(b) - f(a) \right\rvert \leq k\,\lvert b - a \rvert$$

The proof takes one line from the equality: $\left\lvert f(b) - f(a) \right\rvert = \left\lvert f'(c) \right\rvert \lvert b - a \rvert \leq k \lvert b - a \rvert$.

Applied to every pair of points of an interval $I$, it says exactly that $f$ is $k$-Lipschitz on $I$ — the bridge to uniform continuity.

The monotonicity theorem, finally proved

The result we have been using since high school without proving it follows immediately from the mean value theorem. Let $f$ be differentiable on an interval $I$, and $x < y$ two points of $I$. There exists $c$ between $x$ and $y$ such that

$$f(y) - f(x) = f'(c)\,(y - x)$$

Since $y - x > 0$, the sign of $f(y) - f(x)$ is that of $f'(c)$. We deduce all three statements at once:

Hypothesis on $I$	Conclusion
$f' \geq 0$	$f$ increasing
$f' > 0$	$f$ strictly increasing
$f' = 0$	$f$ constant

The last point assumes $I$ is an interval: on $\mathbb{R} \setminus \{0\}$, the function equal to $0$ on $\left]-\infty \,;\, 0\right[$ and $1$ on $\left]0 \,;\, +\infty\right[$ has zero derivative without being constant.

Numerical bounds

The inequality provides bounds without computing a single exact value.

Since $\lvert \cos t \rvert \leq 1$, we obtain for all real numbers $x$ and $y$:

$$\left\lvert \sin x - \sin y \right\rvert \leq \lvert x - y \rvert$$

Likewise, on $[0 \,;\, +\infty\left[\right.$, the function $x \mapsto \sqrt{1 + x}$ has derivative $\dfrac{1}{2\sqrt{1+x}} \leq \dfrac{1}{2}$, hence the classic bound

$$\sqrt{1 + x} \leq 1 + \frac{x}{2}$$

It is also the mean value inequality that bounds the error of numerical methods: a function whose derivative is bounded by $k$ cannot drift from its tabulated value faster than $k$ times the step.

Classic pitfall

Believing that the bound $k$ must be attained. The inequality remains true for any upper bound of $\lvert f' \rvert$: the sharper it is, the better the estimate, but a crude upper bound never makes the statement false. On the other hand, a very real mistake is applying the inequality on an interval where $f'$ is not bounded, such as $x \mapsto \sqrt{x}$ near $0$.