(adsbygoogle = window.adsbygoogle || []).push({}); Lower Semicontinuity

I found this in the web:

We say that [tex]f[/tex] is lower semi-continuous at [tex]x_0[/tex] if for every [tex]\epsilon > 0[/tex] there exists a neighborhood [tex]U[/tex] of [tex]x_0[/tex] such that [tex]f(x) > f(x_0) - \epsilon[/tex] for all [tex]x[/tex] in [tex]U[/tex]. Equivalently, this can be expressed as

[tex]\liminf_{x \to x_0} f(x) \geq f(x_0).[/tex]

The first definition is quite clear to me (by looking at an example of lower semicontinuity diagram). But I don't understand its equivalence to the second definition. Could someone draw the connection?

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# Lowe Semicontinuity

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