Give an example of such f(x) and prove, why it

Lainey Goodwin

Lainey Goodwin

Answered question

2022-01-23

Give an example of such f(x) and prove, why it works
A non constant function f:[0,1]R such that f(x)0 for any x[0,1], but 01f(x)dx=0

Answer & Explanation

Jason Olsen

Jason Olsen

Beginner2022-01-24Added 14 answers

01f(x)dx=0
And we know that abf(x)dx=abf(x)dx+abf(x)dx, acb
Now, let f(x)={0,0x11,x=1
Thus, f(x) is non-constant and f(x)0x[0,1]
Now, 01f(x)dx=01ξf(x)dx+1ξ1f(x)dx, where ξ0
i.e. 01f(x)dx=limξ0(01ξf(x)dx+1ξ1f(x)dx)
=limξ0(01ξ0dx+1ξ11dx)
=limξ0(0+x1ξ1)=limξ0ξ=0
So, 01f(x)dx=0

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