Integrability- Understanding When Functions Can Be Integrated

What Is Integrability, Exactly?

Integrability answers one basic question: can you find the area under this function's curve? That's it. If yes, the function is integrable. If no, it isn't.

Most functions you'll encounter in undergraduate math are integrable. The ones that aren't exist, but they're pathological—built specifically to break rules. You'll probably never stumble onto one by accident.

The Two Main Types of Integration You Need to Know

Riemann Integration

This is what you learned in calculus class. You slice the area under a curve into rectangles, add them up, and let the width of those rectangles shrink toward zero.

Riemann integrable functions must be bounded on a closed interval. They also need to avoid wild oscillations that make the "rectangle approximation" fall apart.

Lebesgue Integration

Lebesgue integration takes a different approach. Instead of slicing vertically, you slice horizontally—grouping points by their function values.

Lebesgue integration handles more functions than Riemann integration. Some functions that fail Riemann's test pass Lebesgue's. But unless you're doing advanced analysis, you won't need this distinction often.

Conditions for Riemann Integrability

For a bounded function on a closed interval, Riemann integrability comes down to one key condition: the set of discontinuities must have measure zero.

In plain English: the function can have isolated jumps and wobbles, but it can't have too many of them packed together.

These functions ARE Riemann integrable:

• Continuous functions on closed intervals
• Functions with finitely many discontinuities
• Monotonic functions (always increasing or always decreasing)
• Step functions
• Piecewise continuous functions

These functions are NOT Riemann integrable:

• Functions with dense discontinuities (like the Dirichlet function)
• Unbounded functions on the interval
• Functions with infinite oscillations at every point

The Dirichlet Function: The Classic Non-Integrable Example

The Dirichlet function defines f(x) = 1 if x is rational, f(x) = 0 if x is irrational.

On any interval, rational and irrational numbers are both dense. No matter how thin you slice it, every rectangle contains both 0-height and 1-height points. The upper sum is always 1, the lower sum is always 0. They never converge.

This function isn't Riemann integrable. But it IS Lebesgue integrable, with an integral of 0.

Thomae's Function: A Surprising Case

Thomae's function (sometimes called the popcorn function) is defined as f(x) = 1/q if x = p/q in lowest terms, and f(x) = 0 if x is irrational.

This function is discontinuous at every rational point. That's dense—infinitely many points. Yet it's Riemann integrable with integral 0.

Why? Because at irrational points, the function equals 0. The set where it differs from 0 has measure zero. The integrability condition only cares about the measure of the discontinuity set, not its cardinality.

How to Check If a Function Is Integrable: A Practical Method

Step 1: Verify Boundedness

The function must be bounded on the closed interval. Check if there's a maximum and minimum value. Unbounded functions fail immediately.

Step 2: Identify Discontinuities

Find where the function isn't continuous. Sketch the graph if needed. Count the jump discontinuities, vertical asymptotes, and oscillations.

Step 3: Assess the Discontinuity Set

Ask: is the set of discontinuities "small" in a measure-theoretic sense?

• Finite discontinuities? Integrable.
• Countably infinite discontinuities? Integrable.
• Dense discontinuities? Usually NOT integrable (unless special structure like Thomae).

Step 4: Apply the Riemann-Lebesgue Criterion

A bounded function on [a,b] is Riemann integrable if and only if the set of points where it's discontinuous has Lebesgue measure zero.

For practical purposes: if the discontinuities form a finite set, a countable set, or a set that "doesn't take up space," you're good.

Comparison: Riemann vs Lebesgue Integrability

Common Mistakes to Avoid

Assuming continuity is required. It's sufficient, not necessary. Functions with isolated discontinuities are still integrable.

Confusing integrability with antiderivatives. A function can be integrable without having an elementary antiderivative. e^(-x²) is integrable everywhere, but you can't write its antiderivative in closed form.

Ignoring the interval. A function might be integrable on [0,1] but not on [0,∞). Integrability is always relative to an interval.

Overthinking pathological cases. If you're calculating areas under reasonable functions (polynomials, trig, exponentials, rational functions with bounded denominators), they're integrable. Save the edge cases for when you actually encounter them.

Quick Reference: Standard Integrable Function Types

• Polynomials — always integrable
• Sine, cosine, and other trig functions — always integrable
• Exponential functions — always integrable
• Rational functions — integrable when denominator has no zeros on the interval
• Continuous combinations of integrable functions — integrable
• Products and quotients (where denominator ≠ 0) of integrable functions — integrable

When You Actually Need Lebesgue Integration

Most engineering, physics, and economics problems use Riemann integration. You'll hit Lebesgue's framework when:

• You're working with random variables and probability theory
• You're dealing with function spaces in functional analysis
• You need rigorous limit theorems under the integral
• You're reading advanced mathematical literature

For everything else, Riemann integration does the job. The theorems are simpler, the intuition is geometric, and the answers are the same for "nice" functions.

The Bottom Line

Integrability isn't complicated once you strip away the jargon. A function is integrable on an interval if you can assign a meaningful area to the region under its curve.

For Riemann integration: bounded function, closed interval, discontinuities that don't crowd together too much. That's the test.

Most functions you care about pass. The ones that fail are constructed specifically to be counterexamples or appear in measure theory courses. You won't accidentally integrate a non-integrable function in a physics problem.