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Let function $$ f\colon \mathbb{R}\to\mathbb{R}$$ is continuous function and satisfies both conditions :

$$ \forall x\in\mathbb{R}, f(f(x))=x \tag{1}$$ $$ \forall x\in\mathbb{R^+}, \int_{-x}^0f(t)dt-\int_0^{x^2}f(t)dt = x^3 \tag{2}$$ where notation $$\mathbb{R^+}$$ is set of positive real numbers.

What I found : $$f(0)=0 \\ \text{f is decrease function} \\ \text{f is symmetry with }y=x $$

How can I find function f which satisfies this conditions?

Thank you for your advice.

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  • $\begingroup$ Hint: What happens when you differentiate the integral equation? $\endgroup$ – Simply Beautiful Art May 9 at 1:58

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