11 10 pts Say whether the statement is true or false a Let 1

11. (10 pts) Say whether the statement is true or false: (a) Let 1 be a sequence whose range has an accumulation point, then this se- quence has a subsequence that converges. 32 (b) The closure of a countable set is also countable. (c) Let f R Rbe a function that is differentiable at the origin, and suppose f(0 0. Then is also differentiable at the origin. (d) Let f la, bl R be a bounded function that is continuous everywhere in its domain except at the points a1,a2 ,aN, then f is integrable on la, (e) Let A and B be two disjoint compact sets of real numbers and let o: inf{Ir yl l z e A, y E B. Then there exists a point zo E A and a point yo E B such that lao yo-o.

Solution

11 (a) We know that A is an accumulation point of sequence {xn} if every neighbourhood of A contains infinitely many points of {xn}. Therefore, we can easily choose a converging subsequence of {xn}. Hence this statement is true.

11 (b) Closure of countable set is not necessarily countable. Therefore, this statement is false.

11 (c) This statement is true because the function does not pass through origin. The only exception of this statement occurs when function passes through origin. In that case, a sharp edge is formed in the function |f| at origin and then |f| is not differentiable. But here, no sharp edge will be formed as the function is not passing through origin, therefore the statement is absolutely true.

11 (d) The function will be Riemann integrable as the summation of areas under the function will be finite and can be approximated.