The uniform limit of a sequence of functions

Question

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.

Answer 1

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.