Description
metric spaces
Just exercise 1,2,5,8
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Explanation & Answer
The solutions are ready. Please read them and ask if something is unclear.It is probably a typo in q.2a.The docx and pdf files are the same.
1. A function 𝑓: 𝑀 → 𝑀1 is called continuous in a point 𝑚 ∈ 𝑀 if it is defined in some
neighborhood of 𝑚 and
∀𝜀 > 0 ∃𝛿(𝜀): ∀𝑥 ∈ 𝐵𝛿(𝜀) (𝑚) ⇒ 𝑓(𝑥) ∈ 𝐵𝜀 (𝑓(𝑚)).
Here 𝐵𝑟 (𝑦) denotes the open ball centered at 𝑦 with the radius 𝑟 in the corresponding
metric space, 𝐵𝑟 (𝑦) = {𝑥 ∈ 𝑀: 𝜌(𝑥, 𝑦) < 𝑟}.
a. It is evident for 𝑐 = 0. For nonzero 𝑐 estimate |𝑐𝑓(𝑥) − 𝑐𝑓(𝑚)| = |𝑐||𝑓(𝑥) − 𝑓(𝑚)|,
𝜀
which is < 𝜀 for ∀𝑥 ∈ 𝐵𝛿(𝜀) (𝑚) (because for those 𝑥 |𝑓(𝑥) − 𝑓(𝑚)| < 𝑐).
𝑐
b. Given 𝜀 > 0 and having 𝛿𝑓 (𝜀) and 𝛿𝑔 (𝜀) from the definition of continuity, use
𝜀
𝜀
𝛿(𝜀) = min (𝛿𝑓 ( ), 𝛿𝑔 ( )) > 0
2
2
which is sufficient:
|(𝑓(𝑥) + 𝑔(𝑥)) − (𝑓(𝑚) + 𝑔(𝑚))| ≤ |𝑓(𝑥) − 𝑓(𝑚)| + |𝑔(𝑥) − 𝑔(𝑚)| <
𝜀 𝜀
+ =𝜀
2 2
for all 𝑥 ∈ 𝐵𝛿 (𝑚).
c. The required estimate is
|(𝑓 ∙ 𝑔)(𝑥) − (𝑓 ∙ 𝑔)(𝑚)| = |𝑓(𝑥)𝑔(𝑥) − 𝑓(𝑥)𝑔(𝑚) + 𝑓(𝑥)𝑔(𝑚) − 𝑓(𝑚)𝑔(𝑚)| =
= |𝑓(𝑥)(𝑔(𝑥) − 𝑔(𝑚)) + (𝑓(𝑥) − 𝑓(𝑚))𝑔(𝑚)| ≤
≤ |𝑓(𝑥)||𝑔(𝑥) − 𝑔(𝑚)| + |𝑔(𝑚)||𝑓(𝑥) − 𝑓(𝑚)|.
The differences |𝑓(𝑥) − 𝑓(𝑚)|, |𝑔(𝑥) − 𝑔(𝑚)| are arbitrary small for 𝑥 close to 𝑚.
The factor |𝑓(𝑥)| is bounded in some neighborhood of 𝑚:
|𝑓(𝑥)| ≤ |𝑓(𝑥) − 𝑓(𝑚)| + |𝑓(𝑚)|,
∀𝑥 ∈ 𝐵𝛿(1) (𝑚) ⇒ |𝑓(𝑥)| ∈ 𝑈1 (𝑓(𝑚)) ⇒ |𝑓(𝑥)| ≤ 1 + |𝑓(𝑚)|.
1
More formally, for a given 1 > 𝜀 > 0 denote 𝑒 = 2(1+|𝑓(𝑚)|+|𝑔(𝑚)|) and pick
𝛿(𝜀) = min (𝛿𝑓 (𝑒), 𝛿𝑔 (𝑒)) > 0. Therefore
|(𝑓 ∙ 𝑔)(𝑥) − (𝑓 ∙ 𝑔)(𝑚)| ≤ |𝑓(𝑥)||𝑔(𝑥) − 𝑔(𝑚)| + |𝑔(𝑚)||𝑓(𝑥) − 𝑓(𝑚)| ≤
≤ (1 + |𝑓(𝑚)|)𝑒 + |𝑔(�...