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Defn: [itex]f: I → ℝ[/itex] is upper semi-continuous at [itex]x_0 \in I[/itex] if [itex]f(x_0) ≥ \limsup{f(x_0)}[/itex].

The book goes on to say that "clearly" this is equivalent to saying,

For any [itex]ε > 0[/itex] there exists a neighborhood [itex]U[/itex] of [itex]x_0[/itex], relative to [itex]I[/itex] such that [itex]f(x) < f(x_0) + ε , \forall x \in U[/itex].

However, this isn't clear to me. Can someone please explain why these statements are equivalent?

The book goes on to say that "clearly" this is equivalent to saying,

For any [itex]ε > 0[/itex] there exists a neighborhood [itex]U[/itex] of [itex]x_0[/itex], relative to [itex]I[/itex] such that [itex]f(x) < f(x_0) + ε , \forall x \in U[/itex].

However, this isn't clear to me. Can someone please explain why these statements are equivalent?

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