14.2.1: SemiContinuities

The lower limit or limit inferior of a sequence $\{x_i\}$ is defined as $$\liminf_{i\rightarrow\infty}x_i=\lim_{i\rightarrow\infty}\left[\inf_{k\geq i} x_k\right]=\sup_{i\geq 0}\left[\inf_{k\geq i} x_k\right].$$ Alternatively, the limit inferior is the smallest cluster point or subsequential limit $$\liminf_{i\rightarrow\infty}x_i=\min\{y\mid x_{i_\nu}\rightarrow y\}.$$ In the same way $$\limsup_{i\rightarrow\infty}x_i=\lim_{i\rightarrow\infty}\left[\sup_{k\geq i} x_k\right]=\inf_{i\geq 0}\left[\sup_{k\geq i} x_k\right].$$ We always have $$\inf_i x_i\leq\liminf_{i\rightarrow\infty} x_i\leq\limsup_{i\rightarrow\infty} x_i\leq\sup_i x_i.$$ Also if $\liminf_{i\rightarrow\infty} x_i=\limsup_{i\rightarrow\infty} x_i$ then $$\lim_{i\rightarrow\infty}x_i=\liminf_{i\rightarrow\infty} x_i=\limsup_{i\rightarrow\infty} x_i.$$

A function is upper semi-continuous at $\overline{x}$ if $$\limsup_{x\rightarrow \overline{x}}f(x)= f(\overline{x}).$$ We have $$\liminf_{x\rightarrow \overline{x}}f(x)\leq\limsup_{x\rightarrow \overline{x}}f(x).$$ A function is continuous at $\overline{x}$ if and only if it is both lower semicontinuous and upper semicontinous, i.e. if $$f(\overline{x})=\lim_{x\rightarrow \overline{x}} f(x)=\liminf_{x\rightarrow \overline{x}}f(x)=\limsup_{x\rightarrow \overline{x}} f(x).$$