In Luenberger Optimization book, pg. 40 upper semicontinuity for a functional is defined as "if given $\epsilon > 0$ there is a $\delta > 0$ s.t. $f(x) - f(x_0) < \epsilon$ for $||x-x_0|| < \delta$". Then it goes on as "a functional is said to be lower semicontinuous at $x_0$ if $-f$ is upper semicontinuous at $x_0$". Functional is continuous if it is both.
However later in the same page Example 1 talks about a functional $f$ defined on $C[0,1]$
$ f(x) = \int_{0}^{1/2}x(t)dt - \int_{1/2}^{1}x(t)dt $
and says "it is easily verified that f is continuous since, in fact, $|f(x)| \le ||x||$".
How did he make this connection? Where did $x_0$ go? And how come there is a relation between the functional and the norm on $x$?
Thanks,