Question

8. (a) Suppose f: R+ R. Show that E = {2 | f(1) = 2019} need not be open or closed in R. (b) Suppose f,g: R+R are continuous. Determine if the set E = {r f(1) = 9 (2) +2019} is open, closed or neither in R.

This is a real analysis question. Please use methods of this sort (or from Ross textbook) to explain. Thanks!

Answer 1

Result 1: Let fi, f: IR R fr hof both be two continuous function continuous function. then are Result 2: Let g: IR YR i continuous function then go (v) is closed in IR where v is closed in IR and g. 1(8) = 3 xEIR: 9, (H) Evi 8)a) Let us define f. 1RR by fa) = 3 2019: if xt(- co, 1] (2020 if xt (1,00) then E= 2x: (fa) = 2019}= f'(2018) which is not closed and not open. and that fg: R R all continuous given by Result 1, ge is continuous and tag2 is also continuous now, E= 2 x P) = x + 2010} = {x: (8-9-) X) = 20193 - (8-9-)-\/2013) Now, we know every singletoul in IR is closed hence 22019} į closed in R hence by Result 2, E is closed in IR and not open (as R is conweted)