Dear all,


It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta\_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta\_x >=f(x)$? I think this is true at all continuous points of $f$. But when $f$ has a jump at $x$, can we properly define this inner product? Does anyone know any references dealing with this matter?


By the way, I checked the wikipedia page about [semicontinuous functions](http://en.wikipedia.org/wiki/Semi-continuity), from where I find Bourbaki's two volumns. But I didn't find any information about such pairing.




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EDIT: 
Following the remark by Tapio Rajala, I think what I want is the following:


Suppose $f$ is a semicontinous function. Then the function 


$$
x\mapsto (f\* \delta\_0 )(x) = \int f(x-y) \delta\_0(y)d y 
$$


is in $L^{\infty}\_{loc}(R)$. 


It seems true for me. If anyone knows a reference, it would be nice, even though the proof seems not difficult. :-)




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Here is another motivation of this problem:


Consider the wave equation in $R$


$$
\frac{\partial^2 }{\partial t^2} u(t,x) = \frac{\partial^2 }{\partial x^2} u(t,x) ,\quad t>0,x\in R\;,
$$ 


with vanishing initial position. Suppose that the initial velocity is a nonnegative Borel measure $\mu$. The solution is


$$
u(t,x) = \mu\left([x-t,x+t]\right) = (\mu \* G\_t)(x)\:,
$$


where $G\_k(x)$ is the fundamental solution:


$$
G\_t(x) = 1\_{|x|\le t}
$$


which is a semicountinuous function. In particular, by letting $\mu=\delta\_0$, we have the problem of pairing a delta function with a semicontinuous function. What I need is simply that the solution $u(t,x)$ is in $L^\infty\_{loc}(R)$. I think this is true.


Thanks a lot for any hints and helps!


RIP, Bill.



Can we calculate the inner product of a semicontinous function with the Dirac delta function?
This is an ill posed problem. The Dirac $\delta$ is a continuous linear map from the (locally convex) space of continuous function on $\mathbb{R}^n$ to $\mathbb{R}$. You ask if it admits an extension to the larger set of semicontinuous functions. First, semicontinuity is not a linear condition: the sum of a lower semicontinuous function with an upper semi-continuous function may not be semi-continuous in any fashion. (Lower semicontinuous functions do form a convex cone.) I assume you want your extension to be linear. So you may be asking for an extension of $\delta$ as a linear function on the larger vector space of functions which can be represented as a sum of a lower semi-continous function with an upper semi-continous function. Extensions of linear functionals from a subspace do exist and by no means are unique. Even requiring some sort of continuity (?) of the extension, the result will not be unique. Without specifying what properties you expect your extension to have you cannot expect a definitive answer. You need to phrase your question more accurately.


