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Let $f:[0,\infty)\to[0,\infty)$ be a differentiable function, and say $\lim\limits_{x\to\infty}f(x)+f'(x)$ exists.
Prove that $f$ is uniform continuous in $[0,\infty)$.

I want to use the fact that if $\lim\limits_{x\to\infty}f(x)$ exists then $f$ is absolutely continuous, but it baffles me.

EDIT: An extreme typo of minus instead of plus, and absolute instead of uniform.

Lior Pollak
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  • Check this: https://math.stackexchange.com/q/407654/42969 – Martin R Feb 08 '22 at 18:01
  • Thanks! Is there a proof that involves the definition of limit by Cauchy directly? – Lior Pollak Feb 08 '22 at 18:04
  • Differentiable functions do not have to be absolutely continuous. Is there an hypothesis missing? Or did you simply want uniform continuity? See: https://math.stackexchange.com/questions/185303/does-the-everywhere-differentiability-of-f-imply-it-is-absolutely-continuous-o – B. S. Thomson Feb 08 '22 at 18:28
  • You are correct, guess I’m super distracted today, sorry – Lior Pollak Feb 08 '22 at 18:40

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