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Integrable Function — Definition, Examples & Properties

Integrable Function

A function for which the definite integral exists. Piecewise continuous functions are integrable, and so are many functions that are not piecewise continuous.

Note: Non-integrable functions are seldom studied in the first two years of calculus.

Key Formula

∫abf(x) dx=lim⁡n→∞∑i=1nf(xi∗) Δx\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x
Where:
  • f(x)f(x) = The function being integrated
  • [a,b][a, b] = The closed interval over which the integral is computed
  • nn = The number of subintervals in the partition
  • xi∗x_i^* = A sample point chosen in the i-th subinterval
  • Δx\Delta x = The width of each subinterval, equal to (b − a)/n

Worked Example

Problem: Determine whether f(x) = x² is integrable on [0, 3], and if so, compute the definite integral.
Step 1: Check integrability. The function f(x) = x² is continuous on the closed interval [0, 3]. Every continuous function on a closed interval is Riemann integrable, so the definite integral exists.
Step 2: Find the antiderivative of f(x) = x².
F(x)=x33F(x) = \frac{x^3}{3}
Step 3: Apply the Fundamental Theorem of Calculus to evaluate the integral.
∫03x2 dx=F(3)−F(0)=273−03=9\int_0^3 x^2\,dx = F(3) - F(0) = \frac{27}{3} - \frac{0}{3} = 9
Answer: f(x) = x² is integrable on [0, 3], and the definite integral equals 9.

Another Example

Problem: Consider the function defined on [0, 1] by f(x) = 1 if x is rational and f(x) = 0 if x is irrational (the Dirichlet function). Is this function Riemann integrable?
Step 1: Consider any partition of [0, 1]. In every subinterval, no matter how small, there are both rational and irrational numbers.
Step 2: For the upper Riemann sum, choose rational sample points in each subinterval. Every term equals 1, so the upper sum equals 1.
Un=∑i=1n1⋅Δx=1U_n = \sum_{i=1}^{n} 1 \cdot \Delta x = 1
Step 3: For the lower Riemann sum, choose irrational sample points. Every term equals 0, so the lower sum equals 0.
Ln=∑i=1n0⋅Δx=0L_n = \sum_{i=1}^{n} 0 \cdot \Delta x = 0
Step 4: Since the upper and lower sums do not converge to the same value regardless of how fine the partition becomes, the Riemann integral does not exist.
Answer: The Dirichlet function is NOT Riemann integrable on [0, 1]. This is one of the classic examples of a non-integrable function.

Frequently Asked Questions

What types of functions are integrable?
All continuous functions on a closed interval [a, b] are Riemann integrable. Functions with a finite number of jump discontinuities (piecewise continuous functions) are also integrable. More broadly, a bounded function on [a, b] is Riemann integrable if and only if its set of discontinuities has 'measure zero,' which informally means the discontinuities are sparse enough not to affect the area calculation.
Can a function with discontinuities be integrable?
Yes. A function can have discontinuities and still be integrable. For example, a step function that jumps at finitely many points is integrable. Even a function with infinitely many discontinuities can be integrable, as long as those discontinuities form a set of measure zero. What prevents integrability is having 'too many' discontinuities, as in the Dirichlet function.

Riemann Integrable vs. Lebesgue Integrable

Riemann integrability, the type encountered in introductory calculus, requires that upper and lower Riemann sums converge to the same limit. Lebesgue integrability, studied in advanced analysis, uses a different approach that measures sets rather than partitioning intervals. Every Riemann integrable function is Lebesgue integrable, but not vice versa — the Dirichlet function, for instance, is Lebesgue integrable (with integral 0) but not Riemann integrable. For standard calculus courses, Riemann integrability is what matters.

Why It Matters

The concept of integrability tells you when you can meaningfully compute the area under a curve or the total accumulation of a quantity. Without knowing a function is integrable, applying the Fundamental Theorem of Calculus or numerical integration methods has no theoretical basis. In applications ranging from physics to probability, confirming integrability ensures that quantities like work, displacement, and expected value are well-defined.

Common Mistakes

Mistake: Assuming a function must be continuous everywhere to be integrable.
Correction: A function can have discontinuities and still be integrable. Piecewise continuous functions (finitely many jumps) are always Riemann integrable. Even some functions with infinitely many discontinuities are integrable, provided the discontinuities form a set of measure zero.
Mistake: Confusing 'having an antiderivative' with 'being integrable.'
Correction: These are different concepts. A function can be integrable without having a nice closed-form antiderivative (e.g., e^{−x²}). Conversely, a function can have an antiderivative at every point yet fail to be Riemann integrable in unusual cases. For most calculus problems the two ideas overlap, but they are not the same thing.

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