I have been asked to prove If $f$ is bounded, then $g(x)= \overline{\lim}_{y\to x} f(y)$ is upper semi continuous.
This means somehow I have to show that for some $x_0$ $$\overline{\lim}_{x\to x_0} g(x)\leq g(x_0)$$ Now $$\overline{\lim}_{x\to x_0}g(x)= \lim_{\delta\to 0}\sup_{0\leq|x-x_0|\leq \delta} g(x)$$ $$=\lim_{\delta\to 0}\sup_{0\leq|x-x_0|\leq \delta}( \overline{\lim}_{y\to x} f(y))$$
So for $g$ to be upper semi continuous, I have to show that
$$\lim_{\delta\to 0}\sup_{0\leq|x-x_0|\leq \delta}( \overline{\lim}_{y\to x} f(y))\leq \overline{\lim}_{y\to x_0} f(y)$$
But how this is true?? Basically my question is why $g$ is upper semi continuous