Question

10. (18 points total: 3 points for each correct answer, 0 points for incorrect answers or no answer Answer "True" or "False” for each of the following: (i) If 8,9:R + R and are both continuous at a number c, then the composition function fog is continuous at c. (ii) If functions hi, h2: R + R and are both uniformly continuous on a non-empty set of real numbers E, then the product h h2 is uniformly continuous on E. (iii) If a function g:R for all 3 ER. + R, then there exists a function G:R + R such that G'(x) = g(x) (iv) If A CB and A is countable, then B is uncountable. (v) If f:R +R is continuous and positive at a number c, then f is differentiable at c. (vi) If K is a non-empty compact set of real numbers, then either K is finite or uncountable.

Answer 1

som (1) hi, h: TRR are both uniformity continuous on non empten set Ert The product ning is not uniformly continuous. Take the)- sin (Vini). It is an uniformly continuons Function. Then take g(x) = x. It is an einn formly contenuous Function over R+ (-E), f(n) g (a) = nsinde is unbounded, and it is easy to check f. g is not uniformly continuous .