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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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    @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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The function $f+g$ is continuous, so $(f+g)^{-1}((-\infty,0])$ is closed. The interval $[a,b]$ is compact. You can write $E = [a,b] \cap (f+g)^{-1}((-\infty,0])$, the intersection of two closed sets, hence $E$ is closed. A closed subset of a compact set is also compact.

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  • $f$ and $g$ are continuous therefore $f+g$ is continuous

  • if $h$ is continuous then $\{x : h(x) \leq 0\}$ is closed

  • $E \subset [a,b]$ therefore it is not unbounded

  • Use Andre Nicolas' comment.