Apostle 4: Limits and Continuity

Question 1

complete(space)

Answer 1

A metric space (S,d) is called complete if every Cauchy sequence in S converges in S. A subset T of S is called complete if the metric subspace (T,d ) is complete

Question 2

Heine theorem

Answer 2

if A is a compact subset of S and f is continuous on A. Then f is uniformly continuous on A

Question 3

Generalized Mean-Valued Theorem

Answer 3

f and g, each have derivative and continuous. Assuming no interior point x at which both df and dg are infinite, then df(g(b)-g(a) = dg(f(b)-g(a))