Restricting us to function of a single real variable, I was used to prove Extreme value theorem via the short way: show that continuous functions preserve compactness, and the job is done.
Now, I want to prove similar results for semicontinuous functions. Wikipedia show me how to do: http://en.wikipedia.org/wiki/Extreme_value_theorem#Extension_to_semi-continuous_functions
This convinces me, however is a bit long and messy.
There is any way to prove the theorem more straightforwardly? For example, proving that an upper semicontinuous function has an image that owns its supremum?