If f is a continuous function from [−a,a] to RR where a in RR with a>0 and if the definite integral from −a to a of f is equal to 0 (with respect to x), then does that mean f must be odd?

princetonaqo3

princetonaqo3

Answered question

2022-10-17

If f is a continuous function from [ a , a ] to R where a R with a>0 and if the definite integral from −a to a of f is equal to 0 (with respect to x), then does that mean f must be odd?
I am struggling to find a counter-example so I tried to see if I could prove this statement but got stuck.
Is the statement true?

Answer & Explanation

Momellaxi

Momellaxi

Beginner2022-10-18Added 14 answers

π π cos x d x = 0
and cos is even.
This is not for a given a, but for all a>0, that we have
a a f ( x ) d x = 0
Differentiate the above w.r.t. a, and you'll obtain f ( a ) + f ( a ) = 0 which means that f is odd.

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