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Suppose I have a sequence of real-numbers $\{a_n\}_n$ and that $\lim_{n \rightarrow \infty}a_n=a$ where $a$ can be finite or infinite. Consider a function $f(\cdot)$ of $a_n$. Under which conditions $$ \lim_{n\rightarrow \infty}f(a_n)=f(\lim_{n\rightarrow \infty}a_n) $$

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Similar questions have been answered here and here but I would like to know whether we can get a similar result without assuming continuity of $f$ and $a$ finite.

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  • Do you take $a_n$ fixed or must $\lim f(a_n)=f(\lim a_n)$ be true for all $a_n$? –  Feb 02 '16 at 16:45
  • Since you ask for a case in which $\lim\limits_{n\to\infty}a_n$ exists but it is not finite, I guess you assume $f$ to have domain $[-\infty,+\infty]$ ? –  Feb 02 '16 at 16:49
  • Short answer is no. This condition is equivalent to continuity of $f $ at $a $. – hardmath Feb 02 '16 at 16:51
  • Hence, imposing continuity of $f(\cdot)$, the result holds also for $a=\infty$ or $a=-\infty$? –  Feb 02 '16 at 19:59

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