Prove that T is closed..

[mikey11]

Suppose f: [a,b] -> R and g: [a,b] ->R are both continuous. Let T = {x: f(x)=g(x)}. Prove that T is close.

Hey i'm new to this site, studying for an upcoming exam and I'm stuck on a few problems in this one specific chapter. Hope some one can help Thanks!

 

[daon]

If f(x)=g(x) for all x in [a,b] then the Img(f) = Img(g) = T. Since [a,b] is a closed interval they attain a maximum and minimum so that the range is bounded and hence the image is closed.

Note X = Img(f) U Img(g) is closed and T C X.If you show X-T is open, you show T is closed (X-T is open => R-T is open => T is closed). So you have a y in X but not T. Try to use continuity to show that there is a disk around y contained in X does not intersect T.