I am trying to understand the definition of lower semi-continuity. A function $f$ is lower semi-continuous at some point $x_0$ if the following holds
$$\lim_{x \to x_0} \inf f(x) \geq f(x_0).$$
This would be an example of a lower semi-continuous function: Lower semi-continuous function
My interpretation of the limit inf would be something like this
$$\lim_{x \to x_0} \inf f(x) = \lim_{n \to x_0} \inf_{x \geq n} f(x).$$
So for the example at hand (1), $\lim_{x \to x_0} \inf f(x) = -\infty$. I don't see how the lim inf is greater than $f(x_0)$.
Have I misunderstood the definition of lim inf and could someone please point me in the right direction? I would like to be able to understand why the criteria for lower semi-continuity holds.
The $\liminf$ and $\limsup$ are a local properties of $f$ around $x_0$ which do not depend on the value of $f$ at $x_0$: $$\lim_{x \to x_0} \inf f(x) = \lim_{r \to 0} \inf_{0<|x-x_0|<r} f(x)$$ and $$\lim_{x \to x_0} \sup f(x) = \lim_{r \to 0} \sup_{0<|x-x_0|<r} f(x).$$
In your picture, for sufficiently small $r>0$, $\inf_{0<|x-x_0|<r} f(x)=f(x_0-r)$ which tends to $f(x_0)$ as $r\to 0$, and therefore $$\lim_{x \to x_0} \inf f(x)=\lim_{r\to 0}f(x_0-r)=f(x_0).$$ On the other hand, you can verify that in your picture, $$\lim_{x \to x_0} \sup f(x)=\lim_{r\to 0}f(x_0+r)>f(x_0).$$