How to define $f(0)$ when $f$ is a function in $L^2$? Any function $f$ in $L^2$ is a actually an equivalence class and has properties that only hold "almost everywhere." But it would be convenient to speak of the value of $f$ at certain points like $f(0)$.  
Is there a meaningful way of defining this?
 A: The more natural way would be to define $f(0)$ as an average of $f$ near 0:
(Re)define $f(0)$ to be the limit as $\varepsilon$ tends to 0 of
$$\frac{1}{\mu(B(0,\varepsilon))}\int\chi_{B(0,\varepsilon)}(x)f(x)d\mu(x).$$
provided this limit exists (and it does a.e. (at least) by the Lebesgue Differentiation Theorem. In this context when this limit exists it is said that 0 is a Lebesgue point, and what the Lebesgue Differentiation theorem says is that almost every point is a Lebesgue point).
This way you're giving $f$ the value you'd give it if you could paint its plot. Think for example in the function $\chi_{\mathbb{Q}\cap[-1,1]}$. However, this won't work if 0 is not a Lebesgue point (take for example the Heavyside function in $[-1,1]$, $\chi_{[0,1]}$)
So this approach won't work in general.
Nevertheless notice that this would not be useful in the theory of $L^p$ functions cause even if you could do this with every point you would end up with a function that would differ from the original one in a set of measure 0, so as a $L^p$ function it would be exactly the same.
A: The problem here is that $\{0\}$ is a set of measure 0 in $\mathbb{R}$, and in general the restrictions to sets of measure 0 are not well defined.  But one can say the following.  If $M^{1/2}$ a subset of $\mathbb{R}^n$ with holder exponent 1/2 boundary, then the 'restriction' operator, defined a priori for functions in $C^\infty(M^{1/2})$, is a continuous map from the sobolev space $H^{1/2}(M^{1/2}) \to L^2(\partial M^{1/2})$ and hence can be extended to all of $H^{1/2}(M^{1/2})$.  Thus when we assume that the functions has half a derivative of regularity, one can define the restriction of a function to $\partial M^{1/2}$  - a set of measure 0 in $M^{1/2}$.  The keyword here is the trace theorem if you want to read more.  I highly recommend Prof. Pierre Germain's notes, or Lions and Magness.
