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

[Dethrone]

Suppose $f$ is a continuous function on $(-\infty,\infty)$. Calculating the following in terms of $f$.

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

Spoiler

Source: Calc I Midterm

 

[Euge]

Spoiler

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 $f(0)$.

 

[Dethrone]

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