Lebesgue's Differentiability Theorem For Monotone Functions

The Differentiability Question for Monotone Functions

A finite monotone function can have jumps, corners, or very steep behavior near particular points. It need not be continuous everywhere, and its difference quotients need not have a finite limit at every point. Nevertheless, on every interval its derivative exists and is finite almost everywhere. The proof separates two issues: interval-covering estimates show that infinite upper derivatives occur only on a null set, while a second covering estimate shows that upper and lower one-sided derivatives cannot disagree except on a null set.

We first work on a nondegenerate compact interval $[a,b]\subset\mathbb{R}$, with Lebesgue measure $m$ on $\mathbb{R}$. The function $f:[a,b]\to\mathbb{R}$ is finite and nondecreasing, meaning $f(x)\le f(y)$ whenever $a\le x\le y\le b$. All Dini derivatives below are defined at interior points $x\in(a,b)$, and may initially take values in $[0,\infty]$.

Definition: For $x\in(a,b)$, the upper and lower right Dini derivatives and upper and lower left Dini derivatives are $$ D^+f(x)=\limsup_{h\downarrow0}\frac{f(x+h)-f(x)}{h},\qquad D_+f(x)=\liminf_{h\downarrow0}\frac{f(x+h)-f(x)}{h}, $$ $$ D^-f(x)=\limsup_{h\downarrow0}\frac{f(x)-f(x-h)}{h},\qquad D_-f(x)=\liminf_{h\downarrow0}\frac{f(x)-f(x-h)}{h}. $$ The limits use positive $h$ small enough that $x+h,x-h\in[a,b]$. Since $f$ is nondecreasing, every quotient here is nonnegative. By the definitions of liminf and limsup, $D_+f(x)\le D^+f(x)$ and $D_-f(x)\le D^-f(x)$.

These same-side inequalities do not assert an ordering between a left derivative and a right derivative. That distinction matters: a monotone function can have a larger left slope than right slope at a corner. The proof uses interval-covering estimates to compare upper and lower derivatives on exceptional sets, not a false pointwise comparison across the two sides.

The Interval-Covering Estimates Used

We use two precise forms of the interval-covering estimates for a finite nondecreasing function on $[a,b]$. These estimates are the standard results from the earlier material on Dini derivatives and covering estimates; their covering arguments are not repeated here. In both statements, $m^*$ denotes Lebesgue outer measure, so the estimates apply whether or not the sets have first been shown measurable.

Interval-covering estimates: Put $M=f(b)-f(a)$. For every $K>0$, $$ m^*\bigl(\{x\in(a,b):D^+f(x)>K\}\bigr)\le \frac{M}{K}, \qquad m^*\bigl(\{x\in(a,b):D^-f(x)>K\}\bigr)\le \frac{M}{K}. $$ Also, for every pair of real numbers $\alpha<\beta$, the set $$ N_{\alpha,\beta} =\{x\in(a,b):\text{some upper Dini derivative at }x\text{ is }> \beta \text{ and some lower Dini derivative at }x\text{ is }<\alpha\} $$ has outer measure zero. Here “upper Dini derivative” means either $D^+f$ or $D^-f$, and “lower Dini derivative” means either $D_+f$ or $D_-f$.

The first estimate controls the size of a set where one of the upper derivatives is large: large difference quotients require enough increase in $f$, and the total increase available is $M$. The second estimate is the key comparison: for fixed separated thresholds $\alpha<\beta$, the points where an upper derivative exceeds $\beta$ while a lower derivative is below $\alpha$ form a null set.

Lebesgue’s Differentiability Theorem

Theorem (Lebesgue’s Differentiability Theorem for Monotone Functions): If $f:[a,b]\to\mathbb{R}$ is finite and nondecreasing, then $f'(x)$ exists as a finite real number for $m$-almost every $x\in(a,b)$. The derivative is the ordinary two-sided derivative at interior points.

Proof. Let $\mathbb{Q}$ denote the rational numbers, and form the exceptional set $$ N=\bigcup_{\substack{\alpha,\beta\in\mathbb{Q}\\ \alpha<\beta}}N_{\alpha,\beta}. $$ There are only countably many rational pairs, and each $N_{\alpha,\beta}$ has outer measure zero by the interval-covering estimate. Countable subadditivity of outer measure therefore gives $m^*(N)=0$.

Fix $x\in(a,b)\setminus N$. We claim that no upper Dini derivative at $x$ is strictly greater than any lower Dini derivative at $x$. Indeed, suppose an upper Dini derivative $U$ and a lower Dini derivative $L$ satisfied $U>L$. Since all four Dini derivatives are nonnegative, there would be rational numbers $\alpha,\beta$ with $L<\alpha<\beta<U$. Then $x\in N_{\alpha,\beta}\subseteq N$, contrary to the choice of $x$. Hence every upper Dini derivative is at most every lower Dini derivative.

In particular, the same-side inequalities from the definition give $$ D^+f(x)\le D_+f(x)\le D^+f(x), \qquad D^-f(x)\le D_-f(x)\le D^-f(x). $$ Thus $D^+f(x)=D_+f(x)$ and $D^-f(x)=D_-f(x)$. The cross-comparisons just proved then force these two common values to agree as well: $D^+f(x)\le D_-f(x)\le D^-f(x)\le D_+f(x)=D^+f(x)$. Write their common value as $d(x)\in[0,\infty]$.

It remains to show that this common value is finite outside a null set. For each positive integer $n$, set $$ S_n=\{x\in(a,b):D^+f(x)>n\}\cup\{x\in(a,b):D^-f(x)>n\}. $$ The first interval-covering estimate and subadditivity imply $m^*(S_n)\le 2M/n$. Let $S_\infty$ be the set where $D^+f(x)=\infty$ or $D^-f(x)=\infty$. Then $S_\infty\subseteq S_n$ for every $n$, so $m^*(S_\infty)\le 2M/n$ for every positive integer $n$. Letting $n$ tend to infinity gives $m^*(S_\infty)=0$. This also covers the case $M=0$; in that case monotonicity forces $f(x)=f(a)=f(b)$ throughout $[a,b]$.

At every $x\notin N\cup S_\infty$, all four Dini derivatives equal the same finite number $d(x)$. The left difference quotients and right difference quotients therefore both converge to $d(x)$ as $h\downarrow0$, which is exactly the existence of the ordinary finite derivative $f'(x)$. Since $N\cup S_\infty$ has outer measure zero, the theorem follows. $\square$

If the domain is any interval $J\subseteq\mathbb{R}$, apply the compact-interval result to a countable collection of compact subintervals whose interiors cover the interior of $J$. The union of their exceptional null sets is still null. The conclusion is thus local and does not require $J$ to be bounded. Endpoints are not included in the assertion about the two-sided derivative.

Worked Example: A Smooth Monotone Function

Take $f(x)=x^2$ on $[0,2]$. This function is finite and nondecreasing because $y^2-x^2=(y-x)(y+x)\ge0$ whenever $0\le x\le y\le2$. For an interior point $x$ and sufficiently small $h>0$, the right quotient is $$ \frac{(x+h)^2-x^2}{h}=\frac{2xh+h^2}{h}=2x+h, $$ and the left quotient is $$ \frac{x^2-(x-h)^2}{h}=\frac{2xh-h^2}{h}=2x-h. $$ As $h\downarrow0$, both tend to $2x$. Thus $D^+f(x)=D_+f(x)=D^-f(x)=D_-f(x)=2x$, which is finite for every $x\in(0,2)$. Check: at $x=1$, taking $h=0.01$ gives right quotient $2.01$ and left quotient $1.99$; both approach $2$, agreeing with the computed derivative.

Worked Example: A Jump with an Infinite One-Sided Dini Derivative

On $[0,1]$, define $f(x)=0$ for $x<\tfrac12$ and $f(x)=1$ for $x\ge\tfrac12$. This is finite and nondecreasing. At every interior point $x\ne\tfrac12$, the function is constant on a sufficiently small neighborhood, so all four Dini derivatives equal $0$. At $c=\tfrac12$, for $0<h<\tfrac12$, the right quotient is $$ \frac{f(c+h)-f(c)}{h}=\frac{1-1}{h}=0, $$ whereas the left quotient is $$ \frac{f(c)-f(c-h)}{h}=\frac{1-0}{h}=\frac1h. $$ Consequently $D^+f(c)=D_+f(c)=0$ and $D^-f(c)=D_-f(c)=\infty$. The ordinary derivative does not exist at $c$, but this single exceptional point has Lebesgue measure zero. Check: for $h=\tfrac14$, the left quotient is $4$, and for $h=\tfrac18$, it is $8$, consistent with divergence to infinity as $h\downarrow0$.

Worked Example: A Corner and the Cross-Side Ordering Pitfall

On $[-1,1]$, let $f(x)=2x$ for $x\le0$ and $f(x)=x$ for $x\ge0$. It is nondecreasing: on each side its slope is positive, and for $x\le0\le y$ we have $f(x)=2x\le0\le y=f(y)$. At $x=0$, for every sufficiently small $h>0$, $$ \frac{f(0)-f(-h)}{h}=\frac{0-(-2h)}{h}=2,\qquad \frac{f(h)-f(0)}{h}=\frac{h-0}{h}=1. $$ Thus $D^-f(0)=D_-f(0)=2$, while $D^+f(0)=D_+f(0)=1$. Check: substituting $h=\tfrac1{10}$ gives left quotient $2$ and right quotient $1$, exactly as the formulas predict.

A wrong argument would claim that every lower Dini derivative is at most every upper Dini derivative, and so would conclude $2\le1$ here. That claim is false: the defining liminf inequality compares quotients from the same side only. The proof of the theorem avoids this error by first showing that, outside the null exceptional sets, no upper Dini derivative can exceed any lower Dini derivative. The corner itself belongs to an exceptional set and causes no conflict with the almost-everywhere conclusion.

What the Theorem Does—and Does Not—Say

The theorem does not claim differentiability at every point, continuity everywhere, or a uniform bound on the derivative. The jump example shows why the exceptional set cannot simply be discarded pointwise. The interval-covering estimates instead show that points with infinite upper derivative, and points with separated upper and lower derivatives, occupy a set of Lebesgue measure zero. Outside that set the left and right difference quotients have the same finite limit.

Monotonicity is essential to the covering estimates: it makes the increments of $f$ nonnegative and limits their total amount on a compact interval to $f(b)-f(a)$. The result is an important step toward the corresponding differentiability theorem for functions of bounded variation, where one combines monotone components and handles their exceptional sets.

Check Your Understanding

In each question, distinguish same-side comparisons from comparisons between left and right Dini derivatives.

For a finite nondecreasing function, which inequalities follow directly from the liminf and limsup definitions?
Explain why the set where $D^+f=\infty$ is contained in $\{D^+f>n\}$ for every positive integer $n$, and how the covering estimate makes this set null.
For the jump function in the worked example, calculate all four Dini derivatives at $x=\tfrac12$.
At the corner in the worked example, which cross-side ordering fails, and why does this not contradict the theorem?
Why does equality of all four Dini derivatives to a finite value imply existence of the ordinary two-sided derivative at an interior point?

Lebesgue's Differentiability Theorem For Monotone Functions

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