Question

Let f:[a,b]→R and g:[a,b]→R be two continuous functions on [a,b]. Show that the set {x∈[a,b]:f(x)=g(x)} is closed in R.

Answer 1

i think we can do this a snap way, although probably can do it directly by showing that the set where \(f\neq g\) is open.

but may be easier to make the new function
\(h:=f-g\) assert \(h\) is continuous and since the inverse image of a closed set is closed we know \(h^{-1}(0)\) is closed as required.