# Show that $\lim_{x\rightarrow a}F(x)$ exists finitely for $\int_a^b|F'(x)|dx<\infty$, $F\in C^1$

Problem. Suppose $$-\infty\le a and $$F:(a,b)\rightarrow R$$ is a $$C^1$$ function such that$$\int_a^b|F'(x)|dx<\infty.$$ Show that $$\lim_{x\rightarrow a}F(x)$$ and $$\lim_{x\rightarrow b}F(x)$$ exist and are finite.

[Hint. Use Dominated Convergence Theorem]

Attempt. For any $$a',b'$$ such that $$a, using the first fundamental theorem of calculus $$\int_{a'}^{b'}F'(x)dx=F(b')-F(a')$$ whenever either side is defined. I want to show that as $$b'\uparrow b$$, $$\int_{a'}^bF'(x)dx+F(a')$$ exists finitely.

How do I proceed?

• Since $\mathbb{R}$ is complete, it suffices to establish the following Cauchy criterion: $$\lim_{s,t\uparrow b}|F(t)-F(s)| = 0.$$ In doing so, note that $$|F(t)-F(s)| = \left|\int_{s}^{t}F'(x)\,\mathrm{d}x\right|\leq\int_{\min\{s,t\}}^{\max\{s,t\}}|F'(x)|\,\mathrm{d}x.$$ Now can you relate this to the Cauchy criterion for the convergence of $\int_{c}^{b^-}|F'(x)|\,\mathrm{d}x$ for some (and in fact, any) $c\in(a, b)$? Apr 20 at 18:28
• @SangchulLee Is this applicable to lebesgue integrals? Could you kindly elaborate/post an answer Apr 20 at 19:14
• Apply the DCT to $F(y) = F(a)+\int_a^b 1_{[a,y]}(x) F'(x) dx$. Similarly for the other. Apr 20 at 19:27
• @copper.hat Can you please post the elaboration? I'm unable to proceed Apr 21 at 7:00
Suppose $$y \in (a,b)$$ and choose $$y' \in (a,y)$$. Then $$F(y) = F(Y') + \int_{y'}^y F'(t)dt = \int 1_{[y',y]}(t) F'(t)dt$$
Let $$y_n \to b$$ (with $$y_n \in (a,b)$$, of course), then since $$|1_{[y',y_n]}(t) F'(t)| \le |F'(t)|$$, the latter is integrable, and $$1_{[y',y_n]}(t) \to 1$$ for all $$t \in [y',b)$$ we see that $$F(y_n) \to F(y')+\int_{y'}^b F'(t)dt$$. Since this is true for any suitable sequence we see that the limit exists.