Is the $ L^{p}$$[0,1]$ norm continuous in p?
Solution 1:
For (2), you've addressed all cases except $f \notin L^p$ with $p < \infty$. As $q \to p^-$, we have $|f|^q \to |f|^p$ pointwise, so by Fatou's lemma $$\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 :)