Real Analysis Answers

Let f and g be continuous functions from a metric space X to metric space Y such that they agree on a dense subset of a X .then prove that f=g

Prove that any uniformly continuous function from a dense subset of a metric space X to a complete metric space Y has a uniformly continuous function from X to Y

For x and y in R, let d(x,y)=(|x-y|)/(1+|x-y|),prove that d defines a.bounded metric on R

Prove that a uniformly continuous function from metric space to another metric space takes cauchy to cauchy sequences

Test the uniform convergence of the series

Σ∞

𝑛=1 x[𝑛/(1+𝑛^2𝑥^2)−(𝑛+1)/(1+(𝑛+1)^2𝑥^2)]

Let Y be a subspace of a metric space (X, d). Then show that 𝐹⊆𝑌 is closed in Y if and only if

𝐹=𝑌∩𝐻 for some closed subset H of X.

If 𝑓 is a continuous function from a metric space X into a metric space Y and {𝑥𝑛} is a sequence in X

which converges to 𝑥 then show that the sequence {𝑓(𝑥𝑛)} converges to 𝑓(𝑥) in Y.

A function is called “piecewise linear” if it is (i) continuous and (ii) its graph consists of finitely many linear segments. Prove that a continuous function on an interval [a, b] is the uniform limit of a sequence of piecewise linear functions. 

Prove that if a series of continuous functions converges uniformly, then the sum function is also continuous. 

Prove or disprove: If f and g are both of bounded variation on [a, b], then so is f · g.