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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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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.

- 4 years, 2 months ago