Another clearly stated (and demonstrated) property of this functional was its lower semi-continuity.

Showing sequential lower semi-continuity is usually the most difficult part when applying the direct method.

Therefore, semi-continuity can be used to prove continuity.

This version is often useful in discussions of semi-continuity which crop up in analysis quite often.

The next section presents two theorems regarding weak sequential lower semi-continuity of functionals of the above type.

In mathematical analysis, semi-continuity (or semicontinuity) is a property of extended real-valued functions that is weaker than continuity.

Lower semi-continuity: if f is lower semi-continuous, then u is lower semi-continuous.

If the continuity of the function f is weakened to semi-continuity, then the corresponding half of the boundedness theorem and the extreme value theorem hold and the values - or + , respectively, from the extended real number line can be allowed as possible values.