Define components in a metric space. What are all the components of the set of all nonzero real numbers under the
1) usual metric on R, and
2) the discrete metric on R?
In a metric space (X, d), a component of a subset A of X is a maximal connected subset of A.
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State the Lebesgue dominated convergence theorem. Find
Prove that the equation (x,y,z) = defines a unique continuously differentiable function of near ,1( − )1 such that ,1(g − )1 = .
A complete metric space is a countable collection of nowhere dense sets.
Any Lebesgue intergrable function is always Riemann integrable.
Define components in a metric space. What are all the components of the set of all nonzero real numbers under the
1) usual metric on R, and
2) the discrete metric on R?
If f: X → Y is a continuous map between metric spaces X and Y and K is a compact subset of X, then show that f(K) is compact.
The function is not uniformly continuous.