Let $f(t)$ be a continuous function on an interval [a,b]. If $\int_a^b |f(t)| dt = 0$, then determine $f(t)$.

Asked May 16 '20 at 12:59

Active May 16 '20 at 16:30

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Let $f(t)$ be a continuous function on an interval [a,b]. If $\int_a^b |f(t)| dt = 0$, then determine $f(t)$.

Here comes my question. With such integrals including absolute values, I know that I can write the given integral as sum of two or more integrals according to the zeros of the function in absolute value. However, I don't know anything about $[a,b]$ if $a<0<b$.

Where should I start?

edited May 16 '20 at 16:30

Masacroso

30,417

asked May 16 '20 at 12:59

yunusemre

This seems to be a duplicate of Why must the integral of a nonzero continuous function be nonzero? – robjohn May 16 '20 at 13:05
Also related to If $f\in \mathcal{R}[a,b]$ satisfies $\int_a^b |f(x)|dx=0$ then $f=0$ almost everywhere. – robjohn May 16 '20 at 13:09

Let $f(t)$ be a continuous function on an interval [a,b]. If $\int_a^b |f(t)| dt = 0$, then determine $f(t)$.

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