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a continuous function

Given a comtnuous function from F:[0,1]→[0,∞) that satisfy the follow:

\(f\left( \frac{1}{2}x+\frac{1}{2} \right)=f\left( x \right)+1\)

\(f\left( 1-x \right)=\dfrac{1}{f\left( x \right)}\)

for all \(x\in (0,1)\)

Determine value of \(\int\limits_{0}^{1}{f\left( x \right)dx}\)

Note by Idham Muqoddas
4 years, 2 months ago

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3 votes

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You need to clarify your definitions a bit. If \(f\) is continuous on \([0,1]\) (including \(0\) and \(1\)) and \[ f\big(\tfrac12x+\tfrac12\big) = f(x) + 1 \qquad 0 < x < 1 \] then this identity is also true when \(x=0,1\), since both sides of this identity are continuous functions on \([0,1]\). In particular, putting \(x=1\), \(f(1) = f(1) + 1\), which is not possible.

Do you mean that \(f\) is only to be defined and continuous on \((0,1)\)? Then \(f\) must be unbounded, tending to \(\infty\) as \(x\to\infty\), so we are looking at either an improper Riemann or a Lebesgue integral.

Mark Hennings - 4 years, 2 months ago

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