Question

Answer ”True” or ”False” for each of the following:

(i) If f,g : R → R and are both continuous at a number c, then the composition function f ◦g is continuous at c.

(ii) If functions h1, h2 : R → R and are both uniformly continuous on a non-empty set of real numbers E, then the product h1h2 is uniformly continuous on E.

(iii) If a function g : R → R, then there exists a function G : R → R such that G0(x) = g(x) for all x ∈ R. (iv) If A ⊂ B 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