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.
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.
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.
$f$ 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.