Let: $f: [a , \infty)\to \mathbb{R}$ such that $\int_a^\infty f$ converge. Prove that $F(x) = \int_a^x f$ is a uniformly continuous

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Can someone help me proving this problem?
Let $f :[a , \infty)\to \mathbb{R}$ such that $\int_a^\infty f$ converge. Prove that $F(x) = \int_a^x f$ is a uniformly continuous function in $[a,\infty)$. Thanks!

score 2 · Answer 1 · edited Apr 13 '17 at 12:20

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Note that $F(x)$ is continuous in $[a,+\infty)$ and the $\lim_{x\to+\infty}F(x)$ exists and it is finite.

Then use Given a continuous function with an asymptote, prove that the function is uniformly continuous.

edited Apr 13 '17 at 12:20

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answered Nov 19 '16 at 10:44

Robert Z

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Let: $f: [a , \infty)\to \mathbb{R}$ such that $\int_a^\infty f$ converge. Prove that $F(x) = \int_a^x f$ is a uniformly continuous

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