Exercise 65Calculus › Single-variable calculus › Integration › Riemann integralHigh schoolAdded 5 Nov 2026

A step function with zero integral

#zero integral#step functions#nonnegative

Statement

Let $f\ge0$ be a step function on $[a,b]$ with $\int_a^bf=0$ . Show that $f$ vanishes at every point of continuity of $f$ .

Hint

If $f$ were positive on some open interval, the integral would be positive.

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Solution

Setup. Suppose $f(x_0)>0$ at a continuity point $x_0$ . By continuity, there is a neighbourhood $]x_0-\varepsilon,x_0+\varepsilon[$ on which $f\ge f(x_0)/2>0$ .

Positive integral. The integral of $f$ over $[a,b]$ is at least $(f(x_0)/2)\cdot2\varepsilon$ if the whole neighbourhood lies in $[a,b]$ , contradicting $\int f=0$ . (If $x_0$ is an endpoint, a similar half-neighbourhood argument applies.)

Conclusion. $f$ vanishes at each continuity point.

A step function with zero integral

Statement

Solution

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