Hey all, having trouble with these two problems





For #1 not really sure how to start, was thinking using the Intermediate Value theorem. The definition of compactness we are using is the Heine-Borel property (any open cover can be reduced to a finite subcover), Bolzano-Weierstrass property, or the set is closed and bounded (I feel like this is the one that will be used).

And for #4 the example of f(x) discussed is

f(x) = (x^2)sin(1/x^2) when x=/=0, 0 when x=0

do u know the open set defn of continuity? if so the first question is fairly easy. just take an open cover of f(a,b]). then using that defn of continuity and that every open cover of [a,b] has a finite subcover, u can show the open cover of f([a,b]) has a finite subcover

Thanks. Yeah that is what I ended up using after reading around on the internet for a bit.

for 1b. closed sets dont get mapped to closed set. for ex take f=e^x then f maps (-infinity,0] to (0,1] which is not closed set
and open sets do not get mapped to open sets. ex f=x^2 then f maps (-1,1) to [0,1) which is not open

The wording on (4) makes me wonder.

If g'(0) = 0 (meaning g is differentiable at 0), then f is differentiable at 0 and f '(0) = 0.

If g'(0) = λ ≠ 0, then g(x) ~ λx at 0, and you can build a function f with |f| ≤ g and f non-differentiable at 0 : take f(x) = g(x)*sin(1/x).

If g is non-differentiable at 0, then just take f = g.

If g(0) ≠ 0, then the condition |f| ≤ g doesn't give anything of value : f can be pretty much anything non-differentiable...

Great, think I got it, thank you for the help