Continuity of the product of two $L^p$ functions

Question

If two functions are in $L^2$, is their product continuous? I am not sure how to connect functions in $L^p$ spaces with continuity?

Answer 1

No, functions in $L^2$ are not necessarily continuous, and their product is not necessarily continuous. The connection with continuous functions is that the space of continuous functions is dense in $L^2$, so that for any $f \in L^2$ and $\epsilon > 0$, there is a continuous function $h$ and a function $g \in L^2$ with small norm $||g||_2 \leq \epsilon$ such that $f = h + g$.

Thus, there is a continuous functions arbitrarily close to any $L^2$ function, close being in the $L^2$ norm.

Answer 2

Not necessarily. By saying a function is in $L^2(\Omega)$ you are just asking that the function $f$ is square integrable in $\Omega$.

What you can say is that, if $ 1\le p < \infty$ then you can be sure you can find a continuous function that is close as much as you want ($\epsilon$-close) to a continuous function. If you want to check it more, just look for continuous functions with compact supports are dense in L^p

Answer 3

No. There exist discontinuous square-integrable functions, e.g. the indicator function of a set of finite measure, e.g. $\mathbb{1}_{[0, 1]}$. Since $\mathbb{1}_{[0, 1]} \cdot \mathbb{1}_{[0, 1]} = \mathbb{1}_{[0, 1]}$ we see that a product of two square-integrable functions may be discontinuous.

There is a subtlety here. The space $L^2(S, \mu)$ on $S$ relative to measure $\mu$ does not strictly speaking consist of functions. Instead, it is defined as the quotient

$$ L^2(S, \mu) = \mathcal{L}^2(S, \mu) / \mathcal{N} $$

where $\mathcal{L}^2(S, \mu)$ is the set of functions on $S$ square-integrable relative to measure $\mu$ and $\mathcal{N}$ is the kernel of the $L^2$ norm. In particular, we identify two square-integrable functions $f$ and $g$ whenever $f(x) = g(x)$ almost everywhere. Thus, the elements of $L^2$ are strictly speaking equivalence classes of functions. A given equivalence class may simultaneously have continuous and discontinuous representatives. For example, the equivalence class of the zero function contains continuous $f(x)=0$ and discontinuous $g(x)=\mathbb{1}_{\{0\}}$.