Homework Help: Semi continuity and Borel Sets - Measurable Functions

1. Oct 6, 2013

SqueeSpleen

In a book I'm reading it says:
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If $f: \mathbb{R} \longrightarrow \mathbb{R}$ is lower semi continous, then $\{f > a \}$ is an open set therefore a borel set. Then all lower semi continous functions are borel functions.
It's stated as an obvious thing but I couldn't prove it.
The definition a lower semi continous function I'm using is:
A function is lower semicontinous in the point $x_0$ if
$\underline{lim}_{x \rightarrow x_{0}} \geq f(x_{0})$
$\underline{lim}_{x \rightarrow x_{0}}= \sup_{\delta > 0 } \{ inf \{ f(x) / |x-x_{0} | < \delta \} \}$
(A function is lower semicontinous in $\mathbb{R}^{p}$ if for all $x \in \mathbb{R}^{p}$ it's lower semicontinous).

Can someone give me a hint please?