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

Thanks a lot for any hints and helps!

RIP, Bill.


  [1]: http://en.wikipedia.org/wiki/Semi-continuity