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Asuume $f$ is LSC(lower semicontinuous) at $x$ . If $t\lt f(x)$, then there exsits $\delta \gt 0$ such that $t\lt f(y)$ for all $y \in B(x,\delta)$. Thus, $t \le \inf\{f(y)| y\in B(x,\delta)\}$ and we conclude that $t$ $\le$ $\lim_{\delta \to 0}$ $\inf\{f(y)| y\in B(x,\delta)\}$. since $t$ is arbirary, $\lim_{\delta \to 0}$ $\inf\{f(y)| y\in B(x,\delta)\} \le f(x)(why??)$

I don' understand how can we conclude final inequality. Please give me a explanation. I was very exhausted understanding above statement.

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I think you got the inequality in the wrong direction, lower semicontinuous should mean \begin{align*} \liminf_{y \to x} f(y) \ge f(x). \end{align*} In the statement, you have shown for all $t < f(x)$, $\liminf_{y \to x} f(y) \ge t$. It should be clear it must be $\liminf_{y \to x} f(y) \ge f(x)$. If you really want details, you can show by contradiction. If $\liminf_{y \to x} f(y) < f(x)$. Let $\delta = f(x) -\liminf_{y \to x} f(y) > 0$. Take $t = f(x) - \delta /2$. Then $f(x) > t$ but this implies $\liminf_{y \to x} f(y) > f(x)$, a contradiction.

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