Question

We say that a real number ? is an isolated point of a set ? if ? is an element of ? and there exists ? > 0

such that ? is the only element of ? that is in the interval (? − ?, ? + ?)

(a) Prove that every element of the set ? = {1,2,3,… } is an isolated point of ?.

(b) Prove that if ? is an isolated point of dom(?), then ? is continuous at a

We say that a real number a is an isolated point of a set S if a is an element of S and there exists 8 >0 such that a is the only element of S that is in the interval (a - 6, a + 8) Prove that every element of the set N 1,2,3, . is an isolated point of N. Prove that if a is an isolated point of domf), then f is continuous at a (a) (b)

Answer 1

4 2) N = { 1, 2, 3 1 2 3 ttt =90 se choose & = 270 ka eh, (a-e, a + E) n A-E, a+E) n 2) & a is, isolated pt of a since as arbitrorry. 2 every element of as is an isolated point of Dr. b) Dololo le let suppose ai dom point of dom (f). let ETO FRON 7870 for 1x-a728 14x) - feas 1 2 18(a) - fea) = ole. 2 is continuous at a