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Which of the following sentences are not always correct?

$A.$ The uniform limit of a sequence of differentiable functions is integrable.

$B.$ The uniform limit of a sequence of integrable functions is integrable.

$C.$ The uniform limit of a sequence of differentiable functions is differentiable.

Martin Sleziak
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    What exactly do you mean by integrable? Having an antiderivative? – Martin Sleziak Jun 26 '14 at 15:44
  • See also: http://math.stackexchange.com/questions/553677/is-there-integrable-function-sequence-which-is-uniformly-converges-to-not-integr, http://math.stackexchange.com/questions/572688/uniformly-convergent-sequence-of-differentiable-functions, http://math.stackexchange.com/questions/153973/sequence-of-differentiable-functions – Martin Sleziak Jun 26 '14 at 15:46

1 Answers1

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None of these are always correct

Counterexample for A: take $f_n : \mathbb{R} \to \mathbb{R}$ to be $$ f_n(x) = 1 $$

Counterexample for B: take $f_n : \mathbb{R} \to \mathbb{R}$ to be $$ f_n(x) = \begin{cases} 1/x & 1 \leq x \leq n\\ 0 & x > n \end{cases} $$

Counterexample for C: take $f_n : \mathbb{R} \to \mathbb{R}$ to be $$ f_n(x) = \sqrt{x^2 + 1/n} $$

A and B are true, however, on compact domains.

Ben Grossmann
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    And (C) remains false on a compact interval of the form $[a,b]$. Just modify the sequence in this answer, considering $f_n,:,\mathbb{R}\to\mathbb{R}$ defined by $$f_n(x) = \sqrt{\left(x-\frac{b-a}{2}\right)^2+\frac{1}{n}}$$ – Nicholas Stull Jun 27 '14 at 00:04
  • I am confused why A is not always true. A sequence of differentiable functions will be continuous and the uniform limit of a sequence of continuous functions is continuous. All continuous functions are integrable, correct? Or is one of my claims true only over compact domains? Also, why is your counter example a true counter-example? Is $1$ not integrable? – Kevin Sheng Sep 16 '15 at 16:47
  • @KevenSheng "integrable" generally means "having a finite integral". So, non-zero constant functions are only integrable on domains of finite measure. Indeed, the statement "all continuous functions are integrable" is not generally true, but is true on compact domains. – Ben Grossmann Sep 17 '15 at 03:22