Fundamental theorem of calculus applied to a continuous function differentiable except on a countable set Consider a function continuous $f : [0, \infty) \to \mathbb{R}$ such that $f$ is differentiable except on a set $S$ containing countably many points, and its derivative on $[0, \infty) \setminus S$ is given by $f'$ (say). Suppose $f' \leq g$, where $\int_0^\infty g$ makes sense. Can we define $\int_0^\infty f'$, and say that $\lim_{t \to \infty} f(t) - f(0) = \int_0^\infty f'$? If not always, are there sufficient conditions on $f$ when we can do it?
 A: I'm not quite sure of the intent of the problem.  Sometimes one might set a problem for a graduate student just to see the reaction.  I've heard of graduate students,
asked on an oral for the value of the Lebesgue integral $\int_0^1 x^2\,dx$, quite baffled.
For one part of the problem you would have known long ago that the condition $f'(x) \leq g(x)$ along with the existence of $\int_0^\infty g(x)\,dx$ does not help to conclude anything  about  $\int_0^\infty f'(x)\,dx$.  Take $g(t)=0$ and $f(t)=-t$ for all $t\geq 0$.  Usually we would have wanted something like $|f'(x) |\leq g(x)$.
Although you were asked this question in the setting of the Lebesgue integral, don't forget what you already know.
[Added note:  the poser of the problem uses the phrase "makes sense" instead of "exists."  So perhaps in this text infinite values are routinely allowed.  In that case consider the $\int_0^\infty g(x)\,dx=+\infty$ case separately and entertain the possibility that $\int_0^\infty f'(x)\,dx=-\infty$ when the former integral is finite.]
The other part of the problem is more sophisticated.  If $f$ is a continuous everywhere differentiable function on $[0,\infty)$ (the countable exceptional set is irrelevant) $f'$ need not be Lebesgue integrable on $[0,T]$ and $f$ need not be absolutely continuous.  But then the condition that $f' \leq g $  for some Lebesgue integrable function $g$ comes to the rescue.
[Note added: if you are assuming only that $\int_0^\infty g(x)\,dx=+\infty$ then there is no reason to imagine that $f'$ is locally integrable.]
This is part of the theory of the Denjoy-Perron (aka Henstock-Kurzweil) integral.  If a DP integrable function on $[0,T]$ is dominated by a Lebesgue integrable function then that function is Lebesgue integrable.  Or one argues that the continuous differentiable function $f$ (or nearly everywhere differentiable function) is ACG$_*$ and then uses the inequality to show that $f$ is absolutely continuous.  Probably not in your course.
So really   your problem is just this:  Suppose that $f$ is a continuous everywhere [or nearly everywhere] differentiable function
on an interval $[a,b]$ and that there is an absolutely continuous function $G$ for which $f' \leq G'$ a.e. Prove that $f$ is absolutely
continuous. 
Focus on that.
[I think the comments about Lipschitz functions and Dini derivatives don't really lead anywhere. The comment about absolute continuity is the key.] 
