# Tagged Questions

Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞, or as both endpoints approach limits.

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### How much can the integrability at zero tell about the decay rate around zero?

Suppose that $g$ is a continuous, nonincreasing and nonnegative function on $(0,1)$. The question is whether one can characterize the integrability of such functions at zero by their decay rates at ...
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### If $\int_2^\infty f(x)^2 dx$ is convergent, is it true that $\int_2^\infty f(x)x^{-3/4} dx$ is convergent?

If $\int_2^\infty f(x)^2 dx$ is convergent, is it true that $\int_2^\infty f(x)x^{-3/4} dx$ is convergent?
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### $\int_{- \infty}^{+ \infty} |f(t)| dt < \infty \implies \int_{-\infty}^{x} f(t) dt$ is continuous?

I've found counter example for $(A),(D)$ and have shown except a bounded interval $F$ is uniformly continuous everywhere else. And so $(B)$ would imply $(C)$ is correct. But I can't show $(B)$ is ...
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