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Let ⃗f :Xn →Yk and⃗g:Yk...

Let ⃗f :Xn →Yk and⃗g:Yk →Zl be two functions between sets Xn ⊆Rn, Yk ⊆Rk, and

Zl ⊆ Rl.

(a) In lecture, I used the fact that if ⃗f is discontinuous on a set of measure zero and ⃗g is continuous, then the composition ⃗g ◦ ⃗f is also discontinuous on a set of measure zero. Prove this in detail.

(b) In this part, we will assume that ⃗g is discontinuous on a set of measure zero and ⃗f is continuous, and again ask whether ⃗g ◦ ⃗f is discontinuous on a set of measure zero. Define ⃗f : [0,1]2 → [0,1]2 and g : [0,1]2 → R by these formulas:

⃗ 1

,x2)

g(y1,y2) =

1 ify1=1andy2isrational 2

0 otherwise

Greg Kuperberg

f(x1,x2) = ( 2

Prove that this is a counterexample, i.e., that ⃗f is continuous, that g is discontinuous on a set of measure zero, but that g ◦ ⃗f is discontinuous on a set that does not have measure zero.

Zl ⊆ Rl.

(a) In lecture, I used the fact that if ⃗f is discontinuous on a set of measure zero and ⃗g is continuous, then the composition ⃗g ◦ ⃗f is also discontinuous on a set of measure zero. Prove this in detail.

(b) In this part, we will assume that ⃗g is discontinuous on a set of measure zero and ⃗f is continuous, and again ask whether ⃗g ◦ ⃗f is discontinuous on a set of measure zero. Define ⃗f : [0,1]2 → [0,1]2 and g : [0,1]2 → R by these formulas:

⃗ 1

,x2)

g(y1,y2) =

1 ify1=1andy2isrational 2

0 otherwise

Greg Kuperberg

f(x1,x2) = ( 2

Prove that this is a counterexample, i.e., that ⃗f is continuous, that g is discontinuous on a set of measure zero, but that g ◦ ⃗f is discontinuous on a set that does not have measure zero.

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