Find a non-negative, non zero function in a specific domain that $\int_{a}^{b}f(x)dx=0.$

Question

I need an integrable function with the domain $[a,b]$, and the function is non-negative, and it is not the zero function in that domain, and $\int_{a}^{b}f(x)dx=0.$ I couldn't find any function like that!!!!

Answer 1

Hints

• $f$ doesn't have to be continuous
• even though $f$ isn't the zero function, it can be zero at some (or even many) points

Answer 2

If a non-negative continuous function $f$ is non-zero at some $x_0 \in [a,b]$, then you can find a neighborhood $B_{\epsilon}(x_0)\cap[a,b]$ around $x_0$ where $f$ is positive. This means that $\int_a^b f d(x) > 0$. So, no continuous function can satisfy your condition.

However, you can create as many discontinuous examples as you want by taking functions of the form:

$$f(x)=\left\{\begin{matrix}1,\textrm{ if x=}x_0,\\0,\textrm{ if x$\neq$}x_0\end{matrix}\right.$$