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, from where I find Bourbaki's two volumns. But I didn't find any information about such pairing.

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 $L^{\infty}(R)$.

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

Thanks a lot for any hints and helps!

RIP, Bill.