Exercise 19Calculus › Single-variable calculus › Integration › Riemann integralHigh schoolAdded 5 Nov 2026

Step functions and their integrals

#step function#partition#integral

Statement

Let $f:[0,3]\to\mathbb{R}$ be defined by
$f(x)=\begin{cases}1&\text{if }0\le x<1,\\ 3&\text{if }1\le x<2,\\ -1&\text{if }2\le x\le3.\end{cases}$
Show that $f$ is a step function and compute $\int_0^3 f(x)\,dx$ .

Hint

A step function is constant on the intervals of a partition. Identify the subdivision and sum $\sum_k c_k(x_k-x_{k-1})$ .

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Solution

Step function. Take the subdivision $\sigma=\{0,1,2,3\}$ . On each open interval $]0,1[$ , $]1,2[$ , $]2,3[$ , $f$ is constant ( $1,3,-1$ respectively). The values at the subdivision points do not matter for the integral, so $f$ is a step function.

Integral.
$\int_0^3 f=\sum_{k=1}^3 c_k(x_k-x_{k-1})=1\cdot1+3\cdot1+(-1)\cdot1=3.$

Conclusion. $f$ is a step function and $\int_0^3 f=3$ .

Step functions and their integrals

Statement

Solution

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