Let [math]f:\mathbb{R} \rightarrow \mathbb{R}[/math] for which is true [math]\displaystyle\int_{0}^{1}\Bigg(\int_{0}^{1}f(x)f(y)dy\Bigg) dx=2\int_{0}^{1}f(x)dx -1[/math].How do I prove that [math]x_{0}[/math] exists,such that: [math]\displaystyle\int_{0}^{x_{0}}f(t)dt=\frac{1}{2}[/math],using calculus theorems?
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