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I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let $f_n$ be a sequence of functions on $\mathbb{R}$ which converges (uniformly or only pointwise) to a limiting function $f$. Which properties of $f_n$ still hold for $f$?

The easy bits are probably: If convergence is uniform, then

-continuity

-boundedness

-Riemann integrability

-analyticity

are preserved as $n\to\infty$ (see, for example, https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch9.pdf and of course http://en.wikipedia.org/wiki/Uniform_convergence).

Do you know others? Are there any properties that are preserved under pointwise convergence? References or (sketches of) proofs would be great, too, if you know any.

Thanks for helping with this list!

Edit: Jochen pointed out in his answer below that in general differentiability does not need to be preserved when taking the limit. However, if $f_n\to f$ pointwise, and $f_n'\to g$ uniformly, then $f'=g$ (see https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch9.pdf). Does this also hold for left- or right-differentiability? If $f_n$ is not differentiable but right-derivatives exist and tend to $g$, is that then the right-derivative of $f$?

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Differentiability is NOT preserved by the limit: Every continuous function is the uniform limit of polynomials.

Point-wise limits of monotone functions are obviously monotone but they needn't be strictly monotone even if so are all $f_n$ and convergence is uniform. The same holds for convex functions.

Borel measurability is stable under point-wise limits.

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  • $\begingroup$ Yes, I was very careless about the differentiability: What one needs is that $f_n\to f$ pointwise, and $f_n'\to g$ uniformly, then $f'=g$. Thanks for your answer! $\endgroup$ – kelu Apr 21 '14 at 0:54

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