There is a theorem in Spivak's Calculus: 
I want to understand the limitations of this theorem, e.g. why it requires uniform convergence and not pointwise convergence. So I have two questions:
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Find a sequence of functions $\{f_n\}$ such that $\forall n: f_n$ is integrable on [a, b], $\{f_n\}$ converges pointwisely to some function $f$ on [a, b], and $f$ is not integrable on [a, b]
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Find a sequence of functions $\{f_n\}$ such that $\forall n: f_n$ is integrable on [a, b], $\{f_n\}$ converges pointwise to some function $f$ on [a, b], $f$ is integrable on [a, b], but $\int_a^b{f} \neq lim_{n \rightarrow \infty}{\int_a^b{f_n}}$
Best Answer
For your first quesion, Carry on Smiling's answer is the same as the one I was going to give. Let $r_1, r_2, \ldots$ be an enumeration of $\mathbb{Q} \cap [0,1]$, and consider the functions $f_n \colon [0, 1] \to \mathbb{R}$ defined by $$ f_n(x) = \begin{cases} 1 & \text{if } x \in \{r_1,\ldots, r_n\} \\ 0 & \text{otherwise.} \end{cases} $$ Each $f_n$ is integrable because it is discontinuous at only a finite number of points. But the pointwise limit $$ f(x) = \begin{cases} 1 & \text{if } x \in \{r_i : i \in \mathbb{N} \} \\ 0 & \text{otherwise.} \end{cases} $$ is not integrable.
For your second question, consider the functions $g_n \colon [0,1] \to \mathbb{R}$ defined by $$ g_n(x) = \begin{cases} n - n^2x & \text{if } 0 < x < 1/n \\ 0 & \text{otherwise}. \end{cases} $$ The region bounded by $g_n$ and the x-axis is a triangle with vertices $(0,0)$, $(0, n)$, and $(1/n, 0)$, which has area $1/2$. Furthermore, $g_n$ converges pointwise to the constant-zero function. We then have $$ \lim_{n \to \infty}\int_0^1 g_n(x)\,dx = \frac{1}{2} \not= 0 = \int_0^10\,dx. $$