Consider the following definitions of the functions of bounded variation.
Defintion 1. $f$ in said to be a function of bounded variation if $f\in L_{loc}^1(\mathbb{R})$ and $\sup\left\{\int\limits_{\Omega}f \operatorname{div}\phi : \phi \in C_c(\mathbb{R}), ||\phi||_{\infty} \leq 1\right\} < +{\infty}$.
Definition 2. Let $ \mathcal{P} =\left\{P=\{ x_0, \dots , x_{n_P}\} \mid P\text{ is a partition of } \mathbb{R} \text{ satisfying } x_i\leq x_{i+1}\text{ for } 0\leq i\leq n_P-1 \right\} $
define $TV(f)=\sup_{P \in \mathcal{P}} \sum_{i=0}^{n_{P}-1} | f(x_{i+1})-f(x_i) |,$ then $f$ is called function of bounded variation if $TV(f)<\infty.$
Clearly definition 1 does not imply definition 2, (for example Dirichlet function). I have the following doubts.
Q1. If $f\in L^1(R)$ is of bounded variation according to definition 2 then for $h\in \mathbb{R}$ $$\int\limits_{R}|f(x+h)-f(x)| \leq C|h|$$ where $C=TV(f).$
[Proof:\begin{eqnarray} \int\limits_{\mathbb{R}} |f(x+h)-f(x)|&=&\sum\limits_{i\in \mathbb{Z}} \int\limits_{(i-1)h}^{ih}|f(x+h)-f(x)|dx \\ &=&\sum\limits_{i\in \mathbb{Z}} \int\limits_0^{h} |f(x+ih)-f(x+(i-1)h))|dx \because \text{change of vairable} \\&=& \int\limits_0^{h} \sum\limits_{i\in \mathbb{Z}}|f(x+ih)-f(x+(i-1)h))|dx \\&\leq& TV(f)|h|.]\end{eqnarray} Does this result hold if $f$ is of bounded variation according to definition 1.?
Q2. Suppose sequence of uniformly bounded functions $\{f_n\}_{n\in \mathbb{N}},$ satisfy $\sup\left\{\int\limits_{\Omega}f_n \operatorname{div}\phi : \phi \in C_c(\mathbb{R}), ||\phi||_{\infty} \leq 1\right\} < C,$ then can we apply Helly's theorem to exctract a subsequence which converges pointwise a.e.? (P.S.: This result is true if $TV(f_n)< C$)
detailed proof/references will be appreciated.
Call $\| f\|_{BV}$ the supremum in definition 1.
Notice that if $f$ is smooth, then $$ \int_{\mathbb{R}} f(x) \phi'(x) \, dx = - \int_\mathbb{R} f'(x)\phi(x)\, dx, $$ and so by the duality between $L^1$ and $L^\infty$, we have $\| f\|_{BV}= \| f'\|_1$ (you can absorb the minus sign into $\phi$).
Now, under the same assumption that $f$ is smooth, by the fundamental theorem of calculus, $$ \int_\mathbb{R}|f(x+h)-f(x)|\, dx = \int_\mathbb{R} \left| \int_0^1 f'(x+th)h\, dt\right| \, dx \leq \int_0^1\int_\mathbb{R} |f'(x+th)|\, dx |h|\, dt = \| f'\|_{1}|h|. $$ This is the inequality in this case. To pass to general $f$ we mollify.
Consider a function $\eta\in C_c^\infty(\mathbb{R})$ supported on the unit ball, nonnegative, even, and $\int_{\mathbb{R}}\eta(x)\, dx =1$. For $\varepsilon>0$ define $\eta_{\varepsilon}(x)=\varepsilon^{-1}\eta(x/\varepsilon)$ and set $g_\varepsilon= g*\eta_\varepsilon$ for any $g\in L^1_{loc}$, i.e. $$ g_\varepsilon(x):= \int_{\mathbb{R}} g(y)\eta_{\varepsilon}(x-y)\, dy. $$ Now fix $\phi\in C_c^\infty(\mathbb{R})$ with $\| \phi\|_\infty\leq 1$, then by Fubini's theorem and using that $\eta_\varepsilon$ is even we have $$ \int_\mathbb{R} f_\varepsilon(x)\phi'(x)\,dx = \int_\mathbb{R}f(x) (\phi')_\varepsilon(x)\, dx = \int_\mathbb{R} f(x) \phi_\varepsilon'(x)\, dx, $$ where the last equality is due to the fact that convolution commutes with derivatives. Since $\| \phi_\varepsilon\|_\infty\leq \| \phi\|_\infty\leq 1$, we see that $\| f_\varepsilon\|_{BV}\leq \| f\|_{BV}$. We're now ready to finish: Fix $f\in L^1(\mathbb{R})$ with $\| f\|_{BV}<\infty$ and consider the $f_\varepsilon$ as above, then $$ \int_{\mathbb{R}}|f_\varepsilon(x+h)-f_\varepsilon(x)|\, dx \leq \| f_\varepsilon\|_{BV}|h| \leq \| f\|_{BV} |h|, $$ and now simply let $\varepsilon\to 0^+$, using that $f_\varepsilon\to f$ in $L^1$ to conclude.
Added: I should also note that (under the $f\in L^1(\mathbb{R})$ background assumption) the two definition are equivalent, if we pick the right representatives (which is obviously necessary, since elements in Def 1 are equivalence classes, while in Def 2 we talk about pointwise values). This is done in G. Leoni's book "A first course in Sobolev spaces". In your example, the Dirichlet function is a.e. equal to a constant, which is clearly of bounded variation no matter the definition.