TITLE: Is the $ L^{p}$$[0,1]$ norm continuous in p?
QUESTION [6 upvotes]: I ran into the following problem when I was doing my homework, and I have no thoughts on where I should start with:
(1) If $f\in L^{2}$, show that $\displaystyle \lim_{p \rightarrow 1^{+}}\int_{[0,1]}|f|^{p}=\int_{[0,1]}|f|$
(2) If $0<p$, show that $\displaystyle \lim_{q\rightarrow p^{-}}||f||_{q}=||f||_{p}$
My first thought was Generalized LDCT, but it didn't seem to work. I also made some other attempts but none of them were successful... Can anybody give me some hints on how I should look at this question?
Also, I know if $p\rightarrow\infty$ then $||f||_{p}\rightarrow||f||_{\infty}$ on $[0,1]$, but does similar continuity in p holds for other $L^{p}[0,1]$ norms in general?
Thank you!
Edit: 
Sorry if I did not make it clear enough in the question. All $L^{p}$ refers to $L^p[0,1]$.
The first question is found here (thanks to t.b.), but the second question remains, mainly because $f$ is not guaranteed to be in any $L^{p}$.

REPLY [2 votes]: For (2), you've addressed all cases except $f \notin L^p$ with $p < \infty$.  As $q \to p^-$, we have $|f|^q \to |f|^p$ pointwise, so by Fatou's lemma
$$\liminf_{q \to p^-} \int |f|^q \ge \int |f|^p = \infty.$$
This means $\int |f|^q \to \infty$ as $q \to p^-$.  Putting the powers of $1/q$ back in is left as an exercise :)