Why an upper-semicontinuous function is defined as : $$ \forall x_0\in \mathbb{R}^n,\ \limsup_{x\rightarrow x_0} f(x) \leq f(x_0) $$ Whereas : $$ \limsup_{x\rightarrow x_0} f(x) = f(x_0) ? $$
In which case could we have : $$ \limsup_{x\rightarrow x_0} f(x) < f(x_0) ? $$ I sense a case where it exists $x_0$ s.t. $f(x_0)=+\infty$ at a point but not completly sure, someone has an idea ?
For instance $f(x)$ defined by $$\begin{array}{rcl} f(1) & = & 1 \\ f(x) & = & 0 \,\,\text{ ef }x\not=1 \end{array}$$ is upper semi-continuous at 1, and we have $$\limsup_{x\rightarrow 1}f(x)=0<f(1).$$