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Assume functions $f$ and $g$ are continuous on the interval $[a,b]$. Show that the set

$E = \{x \in [a,b] : f(x) + g(x) \leq 0 \}$ is compact.

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    Closed and bounded.2012-04-02
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    @AndréNicolas: Why?2012-04-02
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    @Salech: It is the preimage of a closed set under a continuous function, therefore it is closed; and it is bounded because it is a subset of $[a,b]$.2012-04-02
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    @SalechAlhasov: Bounded is built in. For the other part, the interval $(-\infty,0]$ is closed. Now use ordinary topological definition of continuity, or prove that under the more elementary definition of continuity, a function is continuous iff the inverse image of any open set is open.2012-04-02

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