MHB How can the integration limit be determined for a continuous function?

AI Thread Summary
The discussion revolves around determining the integration limit for a continuous function, specifically analyzing the limit of a nested integral involving the function f. The problem presented involves calculating the limit as x approaches 0 of f applied to an integral of f. Participants express appreciation for the solutions provided, highlighting the complexity of the problem. The conversation emphasizes the challenge of finding alternative methods to solve this limit. Overall, the thread showcases a mathematical exploration of continuous functions and their integrals.
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Suppose $f$ is a continuous function on $(-\infty,\infty)$. Calculating the following in terms of $f$.

$$\lim_{{x}\to{0}}f\left(\int_{0}^{\int_{0}^{x}f(y) \,dy} f(t)\,dt\right)$$

Source: Calc I Midterm
 
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Let $g(x) = \int_0^x f(t)\, dt$. Since $f$ is continuous, so is $g$. Therefore, the composition $f\circ g \circ g$ is continuous. We are considering the limit $\lim_{x\to 0} f(g(g(x)))$, which equals $f(g(g(0)))$, by continuity of $f\circ g \circ g$. Since $g(0) = 0$, the limit is $f(0)$.
 
Excellent solution, Euge. Thanks for participating!
I thought this would be an interesting problem, as any other approach to this would be very difficult (if even possible). :D
 
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