Continuity
A function f: X \to \mathbb{R}X$ is lower semi-continuous (l.s.c.) if
f(x) \leq \lim \sup_\limits{ x \to \infty }
\begin{document}
\begin{tikzpicture}[scale=4]
% Axes
\draw[->] (-1,0) -- (1.1,0) node[below] {$x$};
\draw[->] (0,-0.2) -- (0,1.2) node[left] {$f(x)$};
% Step‐function: f(x)=0 for x≤0, f(x)=1 for x>0
\draw[thick] (-1,0) -- (0,0);
\draw[thick] (0,1) -- (1,1);
\fill (0,0) circle (0.8pt) node[below right] {$f(0)=0$};
\draw (0,1) circle (0.8pt);
% Sequence x_n = 1/n, f(x_n)=1
\foreach \n in {1,2,3,4,5,6} {
\node at ({1/\n},1) [orange] {$\times$};
}
% Annotations
\node at (0.7,1.05) {$f(x_n)=1$};
\node at (0.3,0.8) {$x_n\to0$};
\draw[->] (0.3,0.8) -- (0.11,1);
\node[align=left] at (-0.5,0.5)
{$\displaystyle \liminf_{n\to\infty}f(x_n)=1$\\[-2pt]$\displaystyle \ge f(0)=0$};
\end{tikzpicture}
\end{document}