So I am working on the following problem and I am not really sure how to go about approaching it. I always seem to get stuck when trying to start a problem involving the extension of a function to the boundary for some reason and so any tips, hints, comments or even similar problems would be greatly appreciated.
Suppose that $f:(0,1)\rightarrow(-\infty,\infty)$ is abolutely continuous on each closed subinterval of $(0,1).$ If $f'\in L^p(0,1)$ for some $1\leq p\leq \infty,$ prove that $f$ can be extended to be bounded and continuous on $[0,1].$
My attempt at a solution is as follows. So since we are on a finite measure space we automatically have that $$f\in L^q \ \forall q\leq p,$$ but most importantly $q=1.$ We also know that $f\in \textrm{AC}(0,1)$ guarantees $f'$ exists $[m]$ a.e.. Now we have by the Fundamental Theorem of Calculus that for any $x\in(0,1).$ $$f(x)=\int_a^x f'(y) dy + f(a),$$ where $a\in(0,1).$ Now this means $$|f(x)|=\left| \int_a^x f'(y) dy + f(a)\right|\leq ||f'||_1+|f(a)|<\infty.$$ I can't see how this gives me what I want though, because I still don't know anything about the endpoints.
Thanks in advance for any assistance.