Classes of Riemann-Integrable Functions

The definition of the Riemann integral requires $U(f,P) - L(f,P) < \varepsilon$ for some partition $P$. Continuous functions pass this test by uniform continuity. But continuity is far from necessary — many functions with jumps or infinitely many oscillations are still integrable. This checkpoint identifies the two most important classes beyond continuous functions, and pinpoints one simple function that is not integrable.

Recap: the Darboux criterion

$f: [a,b] \to \mathbb{R}$ bounded is Riemann integrable iff for every $\varepsilon > 0$ there exists a partition $P$ with

$$U(f, P) - L(f, P) < \varepsilon.$$

The key quantity is the oscillation on each subinterval: $\omega_i \coloneqq M_i - m_i \ge 0$, so $U - L = \sum \omega_i \Delta x_i$. To make $U - L$ small you need to make either the oscillations or the subinterval widths small.

Monotonic functions are integrable

Theorem. If $f: [a,b] \to \mathbb{R}$ is monotone (increasing or decreasing), then $f$ is Riemann integrable.

Proof (for $f$ increasing). On any subinterval $[x_{i-1}, x_i]$, the supremum and infimum are just the endpoint values:

$$M_i = f(x_i), \qquad m_i = f(x_{i-1}), \qquad \omega_i = f(x_i) - f(x_{i-1}).$$

For the uniform partition $P_n$ with $\Delta x_i = (b-a)/n$:

$$U(f, P_n) - L(f, P_n) = \frac{b-a}{n} \sum_{i=1}^n \bigl(f(x_i) - f(x_{i-1})\bigr) = \frac{(b-a)(f(b) - f(a))}{n}.$$

This telescopes to a single factor, and choosing $n > (b-a)(f(b)-f(a))/\varepsilon$ makes it less than $\varepsilon$. $\square$

The bound does not depend on how $f$ is monotone — it could have infinitely many jump discontinuities — only on its total variation $f(b) - f(a)$.

Piecewise-continuous functions are integrable

A function $f$ is piecewise continuous on $[a,b]$ if there are finitely many points $a = c_0 < c_1 < \cdots < c_k = b$ such that $f$ is continuous on each open interval $(c_{j-1}, c_j)$ and has finite one-sided limits at each $c_j$.

Theorem. Every piecewise-continuous bounded function on $[a,b]$ is Riemann integrable.

Proof sketch. Include each discontinuity point $c_j$ in the partition. On subintervals not containing a $c_j$, $f$ is continuous, so the oscillation can be made arbitrarily small by refining. Near each $c_j$, enclose it in a tiny interval of width $\delta$; the contribution to $U - L$ from those intervals is at most $2\|f\|_\infty \cdot k\delta$, which goes to zero as $\delta \to 0$. Combining gives the Darboux criterion. $\square$

The Dirichlet function is not integrable

Define the Dirichlet function:

$$D(x) \coloneqq \begin{cases} 1 & x \in \mathbb{Q}, \\ 0 & x \notin \mathbb{Q}. \end{cases}$$

$D$ is bounded on $[0,1]$, but it is not Riemann integrable. On every subinterval $[x_{i-1}, x_i]$, both rationals and irrationals are dense, so $M_i = 1$ and $m_i = 0$ for every $i$ and every partition $P$. Therefore

$$U(D, P) = \sum_i 1 \cdot \Delta x_i = 1, \qquad L(D, P) = \sum_i 0 \cdot \Delta x_i = 0,$$

for every partition. The upper and lower integrals are $1$ and $0$ respectively — they never agree. So $D$ is not Riemann integrable.

Lebesgue’s criterion

The precise characterisation of Riemann-integrable functions uses the concept of a set of measure zero: a set $E \subseteq [a,b]$ has measure zero if for every $\varepsilon > 0$ it can be covered by countably many intervals whose total length is less than $\varepsilon$.

Lebesgue’s theorem. A bounded function $f: [a,b] \to \mathbb{R}$ is Riemann integrable if and only if the set of its discontinuities has measure zero.

The proof is beyond the current prerequisites, but the statement unifies the examples above:

• Continuous functions: no discontinuities, measure zero trivially.
• Monotonic functions: at most countably many jump discontinuities, hence measure zero.
• Piecewise-continuous functions: finitely many discontinuities, measure zero.
• Dirichlet function: discontinuous everywhere, so the discontinuity set $[0,1]$ has measure $1 \neq 0$.

Summary

• Darboux criterion: $f$ is integrable iff oscillations $\omega_i = M_i - m_i$ can be made uniformly small.
• Monotonic functions are integrable: the telescoping bound $(b-a)(f(b)-f(a))/n \to 0$ regardless of how many jumps $f$ has.
• Piecewise-continuous functions are integrable: isolate discontinuities in tiny intervals; continuity handles the rest.
• The Dirichlet function $D = \mathbf{1}_\mathbb{Q}$ is not integrable: upper sum is always $1$, lower sum always $0$.
• Lebesgue’s criterion: a bounded function is Riemann integrable iff its set of discontinuities has measure zero.